Exponential functions ☰

Exponential functions are a type of mathematical function that involves a constant base raised to a variable exponent. The general form of an exponential function is: \(f(x)=a^x\), where a is a positive constant called the base, and \(x\) is the variable exponent. Exponential functions have a distinctive shape: as \(x\) increases, the function grows or decays at an increasing rate, depending on whether \(a > 1 \) or \(0 < a < 1 \). In other words, the function increases or decreases very rapidly as \(x\) moves away from zero.

One of the most well-known exponential functions is the natural exponential function, which has base \(e\), a mathematical constant approximately equal to \(2.71828\). The natural exponential function is denoted by: \(f(x)=e^x\).

The natural exponential function has many important applications in mathematics, science, and engineering. For example, it appears frequently in calculus, differential equations, and probability theory.

Exponential functions also exhibit some important properties that make them useful in modeling real-world phenomena. One of these properties is that the product of two exponential functions with the same base is itself an exponential function with the same base, but with an exponent equal to the sum of the original exponents. That is, for any positive constants \(a\): \(a^x \cdot a^y =a^{x+y} \).

Another important property of exponential functions is that the ratio of two exponential functions with the same base is itself an exponential function with the same base, but with an exponent equal to the difference of the original exponents. That is, for any positive constants \(a\): \( \frac{a^x}{a^y} =a^{x-y} \).

Exponential functions also have an inverse function called the natural logarithm, denoted by \(ln(x)\). The natural logarithm is defined as the inverse of the natural exponential function. That is, for any positive real number \(x\): \( ln(e^x )=x \).

The natural logarithm has many useful properties, including the fact that it is the only logarithmic function that is continuous and differentiable on its domain. Additionally, the natural logarithm has an important relationship with calculus, since its derivative is equal to the reciprocal of its argument: \( \frac{d}{dx} ln(x) = \frac{1}{x} \).

Exponential functions and their properties play an important role in many areas of mathematics and science, including finance, population growth, radioactive decay, and electrical circuits, to name just a few. By understanding exponential functions and their behavior, mathematicians and scientists can better model and understand complex phenomena in the natural world.

Exponential growth and decay: As mentioned earlier, exponential functions exhibit rapid growth or decay as the variable exponent increases or decreases. When the base a is greater than 1, the function grows exponentially and is called an exponential growth function. On the other hand, when the base a is between 0 and 1, the function decays exponentially and is called an exponential decay function. The rate of growth or decay is proportional to the size of the function at any given time or point.

Exponential functions and calculus: Exponential functions play an important role in calculus, particularly in the context of differentiation and integration. The derivative of an exponential function with base a is given by: \(\frac{d}{dx} a^x=a^x ln(a) \).

This means that the rate of change of an exponential function is proportional to the function itself. Moreover, the integral of an exponential function can be computed using the formula:
\(\int a^x \) , \(dx =\frac{a^x}{ln(a)} +C \), where \(C\) is the constant of integration. This formula allows us to compute the area under an exponential curve or the total growth or decay over a certain time period.

Applications of exponential functions: Exponential functions are used in a wide range of applications in fields such as finance, biology, physics, chemistry, and engineering.

• In finance, exponential functions are used to model compound interest and investment growth.
• In biology, exponential functions are used to model population growth and bacterial growth.
• In physics, exponential functions are used to model radioactive decay and electrical circuits.
• In chemistry, exponential functions are used to model chemical reactions and enzyme kinetics.
• In engineering, exponential functions are used to model signal attenuation and filter response.

Complex exponential functions: In addition to real exponential functions, there are also complex exponential functions, which involve a complex base raised to a complex exponent.
The most common complex exponential function is the complex exponential with base \(e\), given by: \( e^{ix}=cos(x)+i sin(x)\), where \(i\) is the imaginary unit and \(cos(x)\) and \(sin(x)\) are the cosine and sine functions, respectively. The complex exponential function has many important applications in mathematics, physics, and engineering, particularly in the context of signal processing, Fourier analysis, and quantum mechanics.

The graph of an exponential function. ☰

The graph of an exponential function is a representation of the function's behavior on a Cartesian coordinate system, using the \(x\)-axis for the input values and the \(y\)-axis for the output values. An exponential function has the form: \(y = ab^x\), where: '\(a\)' is a constant, called the initial value or amplitude, '\(b\)' is the base, which determines the growth rate of the function, '\(x\)' is the input variable, and '\(y\)' is the output value.

Exponential functions can be divided into two categories: exponential growth and exponential decay.

1. Exponential Growth:
If the base \((b)\) is greater than 1, the function represents exponential growth. In this case, the graph increases as \(x\) increases. Key features of an exponential growth graph include:

• The function passes through the point \((0,a)\), as any value raised to the power of 0 is 1.
• The \(y\)-intercept is at the point \((0,a)\).
• As \(x\) approaches negative infinity, \(y\) approaches 0, but never touches the \(x\)-axis. This creates a horizontal asymptote at \(y=0\).
• As \(x\) increases, the graph becomes steeper and rises more quickly.

2. Exponential Decay:
If the base \((b)\) is between 0 and 1, the function represents exponential decay. In this case, the graph decreases as \(x\) increases. Key features of an exponential decay graph include:

• The function passes through the point \( (0,a) \), as any value raised to the power of 0 is 1.
• The \(y\)-intercept is at the point \((0,a)\).
• As \(x\) approaches negative infinity, the graph rises more quickly.
• As \(x\) approaches positive infinity, \(y\) approaches 0, but never touches the \(x\)-axis. This creates a horizontal asymptote at \(y=0\).

Understanding the behavior of exponential functions and their graphs is important in various fields, such as finance, biology, and engineering, as they often model natural processes like population growth, radioactive decay, and compound interest.

Value of e☰

The value of '\(e\)' is a mathematical constant that is approximately equal to \(2.718281828459045\). It is an irrational number, which means its decimal representation neither repeats nor terminates. The constant '\(e\)' is named after the Swiss mathematician Leonhard Euler, although the number was discovered by Jacob Bernoulli while studying compound interest problems.

'\(e\)' has many important properties and applications in various branches of mathematics, including calculus, number theory, and complex analysis. Some key aspects and applications of '\(e\)' include:

Exponential Functions:
'\(e\)' is the base of the natural exponential function, which is written as \(y=e^x\). This function has the unique property that its slope (derivative) at any point is equal to its value at that point. This property makes the natural exponential function essential in solving differential equations and modeling growth and decay processes.

Natural Logarithm:
The natural logarithm, denoted as \(ln(x)\), is the logarithm with base '\(e\)'. It is the inverse of the natural exponential function. In other words, if \(y=e^x\), then \(x=ln(y)\). The natural logarithm plays a crucial role in calculus, especially when dealing with integration and differentiation of exponential functions.

Compound Interest:
The constant '\(e\)' first arose in the context of compound interest problems. The formula for compound interest is given by: \( A = P(1 +\frac{r}{n} )^{nt} \), where: \(A\) is the final amount, \(P\) is the principal (initial amount), \(r\) is the interest rate (as a decimal), \(n\) is the number of times interest is compounded per time period, \(t\) is the number of time periods.
As the frequency of compounding \((n)\) approaches infinity, the formula converges to the continuous compound interest formula: \( A = P \cdot e^{rt} \)

Euler's Identity:
'\(e\)' is a key component of Euler's identity, which is considered one of the most beautiful and profound equations in mathematics. Euler's identity is given by: \( e^{i \pi}+1=0 \).
This equation connects five fundamental constants in mathematics: '\(e\)', '\(i\)' (the imaginary unit), \( \pi \) (pi), 1, and 0.

Taylor Series:
'\(e^x\)' has a simple and convergent Taylor series expansion around \(x=0\), which is given by:
\( \small e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \ldots = \sum_{}^{} \frac{x^n}{n!} \)

This infinite sum converges for all values of \(x\) and provides a way to approximate the value of \(e^x\) for any given \(x\).

Factorial Function Approximation:
'\(e\)' is involved in approximating the factorial function using Stirling's formula. Stirling's formula is an approximation for the factorial of a large number '\(n\)' and is given by: \( n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n \).
Stirling's formula is useful in various applications, such as combinatorics, probability, and statistical mechanics.

Exponential Integral Function:
The exponential integral function, denoted by \(Ei(x)\), is a special function defined as the following improper integral: \(Ei(x)= \int \left(\frac{e^t}{t} \right) dt \) from \(-x\) to \( \infty \).

This function has several applications in engineering, physics, and applied mathematics, such as the study of electrical circuits, heat conduction, and fluid dynamics.

Chaos Theory and Transcendental Dynamics:
'\(e\)' also appears in the study of chaotic and complex dynamical systems. The constant appears in various contexts, such as the analysis of fractals and the study of the convergence properties of certain iterative algorithms.

Laplace Transforms:
'\(e\)' is used in the Laplace transform, which is a powerful technique for solving linear differential equations by transforming them into algebraic equations. The Laplace transform of a function \(f(t)\) is given by: \( L{f(t)}=F(s)= \int \left(e^{-st} \cdot f(t) \right)dt \) from \(0\) to \( \infty \).

The Laplace transform and its inverse play crucial roles in engineering, physics, and applied mathematics, particularly in the analysis of signals and systems.

In summary, the value of '\(e\)' is a fundamental mathematical constant with numerous applications and properties, making it an essential topic in mathematics and its related fields.

Logarithm ☰

A logarithm is a mathematical concept that allows us to find the exponent to which a certain base must be raised to obtain a given value. Logarithms are the inverse operation of exponentiation, which means that they reverse the process of raising a base to a power.

The logarithm function can be represented as: \( \log_b a=x \).
In this equation, \(b\) is the base of the logarithm, \(a\) is the number we want to find the logarithm of, and \(x\) is the exponent to which we must raise the base \(b\) to obtain \(a\). In other words: \(b^x=a\).

There are two common bases for logarithms: the natural logarithm and the common logarithm.

1. Natural logarithm: The natural logarithm has a base of \(e\) (Euler's number, approximately equal to \(2.71828\) ). The natural logarithm is denoted by the symbol \(ln\), so: \(ln=\log_e a \).

2. Common logarithm: The common logarithm has a base of 10 and is often used in scientific calculations and logarithmic scales. It is usually denoted as log, without any subscript: \( loga=\log_{10} a \).

Logarithms have several important properties that make them useful in a variety of mathematical applications:

• Product rule: The logarithm of a product is the sum of the logarithms of the individual factors.
  \(\log_b (a \cdot c)=\log_b a+\log_b c \).
• Quotient rule: The logarithm of a quotient is the difference between the logarithms of the numerator and the denominator.
  \(\log_b \frac{a}{c} = \log_b a -\log_b c \).
• Power rule: The logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the base number.
  \(\log_b a^c = c \cdot \log_b a \).
• Change of base formula: This formula allows us to convert between logarithms of different bases.
  \(\log_b a=\frac{\log_c a}{ \log_c b} \).
• Identity property: The logarithm of 1 with any positive base \(b\) is always \(0\), and the logarithm of the base with itself is always \(1\).
  \(\log_b 1 = 0 \) and \(\log_b b=1 \).
• Inverse property: If the logarithm and exponentiation operations are applied sequentially with the same base, they will cancel each other out.
  \(b^{\log_b a} = a \) and \(\log_b b^a=a \).
• Logarithmic equations: Logarithms can be used to solve equations involving exponential functions. For example, consider the equation: \( b^x=a \).
  
  To solve for \(x\), we can take the logarithm of both sides: \(\log_b b^x=\log_b a \).
  Using the inverse property, we have: \( x =\log_b a \).

Applications:

• Compound interest: Logarithms can be used to calculate the time required to reach a certain amount of money in a bank account, given an initial deposit and a fixed annual interest rate.
• Earthquake intensity: The Richter scale measures the intensity of earthquakes using logarithms. The scale is logarithmic because the amplitude of seismic waves can vary over a wide range.
• Acoustic intensity: The decibel scale, used to measure sound intensity, is based on logarithms. This scale is convenient because human perception of sound intensity is approximately logarithmic.
• Information theory: Logarithms play a key role in measuring information content, entropy, and the efficiency of data compression algorithms.
• Signal processing: Logarithms are used in Fourier analysis, Laplace transforms, and other signal processing techniques to simplify calculations and analyze the frequency content of signals.
• Population growth: Logarithmic and exponential functions are used to model population growth, radioactive decay, and other natural phenomena that exhibit exponential behavior.
• Computer science: Logarithms are used in algorithms and data structures like binary search, divide-and-conquer algorithms, and the analysis of algorithm complexity.

These are just a few examples of how logarithms are used in various fields. Overall, logarithms play a vital role in simplifying calculations, solving problems, and understanding the behavior of various natural and artificial systems.

Logarithmic function ☰

A logarithmic function is a function that involves the logarithm of a variable or expression. Logarithmic functions are the inverse of exponential functions, meaning that they reverse the process of raising a base to a power. The general form of a logarithmic function is: \(f(x)=\log_b x \).

In this equation, \(f(x)\) represents the logarithmic function, \(b\) is the base of the logarithm, and \(x\) is the input variable or argument. The base \(b\) must be a positive number different from 1. The domain of the logarithmic function is \( (0, \infty ) \), meaning that logarithmic functions are only defined for positive input values.

There are two common logarithmic functions:

1. Natural logarithmic function: The natural logarithmic function has a base of \(e\) (Euler's number, approximately equal to \(2.71828 \) ). The natural logarithmic function is denoted as: \( f(x)=\ln x=\log_e x\).

2. Common logarithmic function: The common logarithmic function has a base of 10 and is often used in scientific calculations and logarithmic scales. The common logarithmic function is denoted as: \( f(x)= \log x=\log_{10} x\).

Properties of logarithmic functions:

• The graph of a logarithmic function is a smooth curve that passes through the point \((1,0)\), since the logarithm of 1 is always 0 for any positive base \(b\).
• The graph of a logarithmic function has a vertical asymptote at \(x=0\), which means that the function approaches infinity as the input value approaches 0.
• The graph of a logarithmic function is symmetric with respect to the line \(y=x\) when compared to the graph of its corresponding exponential function, \(y=b^x\).
• Logarithmic functions are increasing functions if the base \(b > 1\), and they are decreasing functions if \(0 < b < 1\).
• The derivative of a logarithmic function is given by:
  a. Natural logarithmic function: \( \frac{d}{dx} (\ln x ) = \frac{1}{x} \).
  b. General logarithmic function: \( \frac{d}{dx} (\log_b x) = \frac{1}{x \ln b} \).

Logarithmic functions are widely used in various fields, including mathematics, physics, engineering, computer science, and economics. They are especially useful for solving exponential equations, simplifying complex expressions, and modeling phenomena that exhibit exponential behavior.

Logarithmic scale ☰

A logarithmic scale is a non-linear scale used to represent data that spans multiple orders of magnitude. In a logarithmic scale, the value of a quantity is represented by the logarithm of the actual value, instead of the value itself. This scaling method is particularly useful when dealing with data that exhibit exponential growth or decay or when the range of values is very large.

The logarithmic scale can be represented using any base, but the most common bases are 10 (common logarithm) and \(e\) (natural logarithm).

For a value \(x\) on a logarithmic scale with base \(b\), the actual value \(y\) can be obtained using the following formula: \(y=b^x\)

Conversely, to convert an actual value \(y\) to its logarithmic representation \(x\) with base \(b\), the formula is: \(x=\log_b y \).

Some examples of logarithmic scales and their applications include:

• Richter scale: The Richter scale is used to measure the magnitude of earthquakes. It is a logarithmic scale with base 10, and each whole number increase on the scale corresponds to a tenfold increase in the amplitude of the seismic waves.
• Decibel scale: The decibel scale measures the intensity of sound or other quantities like power, voltage, or current, in terms of logarithmic ratios. In the case of sound intensity, the decibel scale is defined as:
  \(L=10 \log_{10} \frac{I}{I_0} \) , where \(L\) is the sound level in decibels, \(I\) is the intensity of the sound, and \(I_0\) is the reference intensity, usually the threshold of human hearing.
• \(pH\) scale: The pH scale is used to measure the acidity or alkalinity of a solution. The scale ranges from 0 to 14, with 7 being neutral. The pH value is defined as the negative logarithm of the hydrogen ion concentration in the solution:
  \( pH =-\log_{10} [H^+] \).
• Logarithmic graph paper: Logarithmic graph paper has one or both axes scaled logarithmically, which allows for the visualization of data that spans several orders of magnitude. This can be especially useful in fields like biology, chemistry, and physics, where quantities can vary greatly.
• Finance and economics: Logarithmic scales are used in finance to represent stock prices, economic indicators, and other variables that may exhibit exponential growth or decay over time.

The logarithmic scale is a powerful tool for representing data and visualizing relationships that might otherwise be difficult to see due to the large range of values. It is widely used in various fields to analyze and communicate information more effectively.

Exponential equations ☰

The exponential equation is a type of equation where the variable appears in the exponent. Exponential functions are widely used in mathematics, science, and engineering to model various phenomena such as population growth, decay, and compound interest.

An exponential function is defined as: \(f(x)=a \cdot b^x \), where: \(a\) is the initial value or the coefficient, which is a non-zero constant. \(b\) is the base, which is a positive constant and different from 1. \(x\) is the exponent, which is the variable.

When solving an exponential equation, the main goal is to isolate the variable \((x)\) to find its value\((s)\). There are several methods to solve exponential equations, including:

Using the properties of exponents:
If you have two equal exponential expressions with the same base, you can set their exponents equal to each other and solve for the variable: \(b^x=b^y \rightarrow x=y\).

Taking the natural logarithm \((\ln) \) or common logarithm \((\log) \) of both sides:
In cases where the bases are different, logarithms can be used to simplify the equation. Using the property of logarithms, you can bring the exponent down and convert the equation into a linear or algebraic equation.

For example, let's solve the equation: \(3^x=9\).
Taking the natural logarithm of both sides: \(\ln(3^x ) = \ln(9) \).
Using the logarithm property, we can bring the exponent down: \(x \ln(3)= \ln9 \).
Solving for \(x\): \( x= \frac{\ln(9)}{\ln(3)} \) .

Using the change of base formula:
If you have an equation of the form: \(a^x=b^y\).
You can use the change of base formula: \(x= \frac{\log_b (a)}{\log_b (b) } \).

Substitution:
In some cases, substitution can be used to simplify the problem. If an exponential expression has a power of another exponential expression, you can substitute the inner exponential expression with a new variable to create a simpler equation to solve.

For example, let's solve the equation: \((2^x )^3=64 \).
Substituting \(y=2^x\), we get: \(y^3 =64 \).
Solving for \(y\), we get \(y=4\). Then, we substitute back \(2^x\) for \(y\): \(2^x=4\).
Solving for \(x\), we find \(x=2\).

Logarithmic equations ☰

Logarithmic equations are mathematical expressions that involve logarithms. A logarithm is the inverse operation of exponentiation and is denoted by the symbol "\(log \)". The logarithm of a number "\(x\)" to the base "\(b\)" is written as \( \log_b x \), which represents the power to which "\(b\)" must be raised to obtain "\(x\)".
There are some important properties of logarithms that are useful in solving logarithmic equations:

\( \log_b (xy) = \log_b x+\log_b y \)

\( \log_b \left(\frac{x}{y} \right) = \log_b x - \log_b y \)

\( \log_b x^y = y \log_b x \)

\( \log_b x = \frac{\log_c x}{\log_c b} \)


The most common bases used in logarithms are base 10 (common logarithm, written as \( \log x) \) and base "\(e\)" (natural logarithm, written as \( \ln x \) ), where "\(e\)" is the Euler's number (approximately \(2.71828\) ).

Now, let's discuss how to solve logarithmic equations. There are three primary methods:

1. Eliminating the logarithm:
If the equation has a single logarithmic expression, we can eliminate the logarithm by using exponentiation.
For example: Given: \( \log_b x=y \). To eliminate the logarithm, we can write this as: \(x=b^y \).

2. Combining logarithmic expressions:
If the equation has multiple logarithmic expressions, we can use the properties of logarithms to combine them into a single expression.
For example, Given: \( \log_b x +\log_b y = \log_b z \)
Using the product rule, we can combine the logarithms: \( \log_b (xy) = \log_b z \).
Now, we can eliminate the logarithm by exponentiation: \(xy=z\).

3. Applying logarithm to an equation: If the equation has no logarithms, but involves exponential expressions, we can apply logarithms to simplify the equation.
For example, Given: \(b^x=y \) To apply the logarithm, we can write this as: \( \log_b (b^x ) = \log_b y \).
Using the power rule, we get: \( x \log_b b = \log_b y \).
Since \( \log_b b=1 \), the equation simplifies to: \( x= \log_b y \).

In summary, logarithmic equations involve mathematical expressions containing logarithms. Solving such equations requires a solid understanding of the properties of logarithms and the appropriate techniques to manipulate and eliminate logarithmic expressions.

Exponential inequality ☰

Exponential inequality is a mathematical concept that arises when you have inequalities involving exponential functions.
An exponential function is a function of the form \(f(x)=a^x\) or \(f(x)=ab^x\), where \(a\), \(b\), and \(x\) are real numbers, and \(b > 0 \), \(b \neq 1\).

An exponential inequality is an inequality where at least one side involves an exponential function. Here are a few examples of exponential inequalities:
1. \( 2^x > 8 \)
2. \( 3^{x-1} \le 27 \)
3. \( 5e^{2x} < 100 \)

To solve exponential inequalities, we often use logarithms, which are the inverse functions of exponential functions. The two most common logarithms are the natural logarithm (denoted as \( \ln \) ) and the common logarithm (denoted as \( \log \) ). The natural logarithm has a base of \(e\), where \( e \approx 2.71828 \), while the common logarithm has a base of 10.

Here's a general approach to solving exponential inequalities:

• Isolate the exponential term on one side of the inequality.
• Apply the logarithm to both sides of the inequality, keeping in mind the properties of logarithms.
• Solve the resulting inequality for \(x\).

Let's illustrate this approach using the first example: \(2^x > 8 \).
The exponential term is already isolated. Now, we apply the logarithm to both sides. We can use any logarithm, but for simplicity, let's use the natural logarithm:
\( \ln (2^x ) > \ln (8) \).

Using the property of logarithms, \( \ln (a^b )= b \ln (a) \), we can simplify the left side of the inequality:
\( x \ln (2) > \ln (8) \).

Now, divide both sides by \( \ln (2) \) to solve for \(x\):
\( x > \frac{\ln (8)}{\ln (2)} \) .

Calculating the numerical values:
\( x > \frac{\ln(2^3 )}{\ln (2)} \).

So the solution to the exponential inequality \( 2^x > 8 \) is \(x > 3\).

Example 2: \( 3^{x-1} \le 27 \)
1. Isolate the exponential term: It's already isolated in this case.
2. Apply the logarithm to both sides: We'll use the natural logarithm for consistency. \( \ln( 3^{x-1} ) \le \ln (27) \).
3. Simplify using the logarithm properties: \( ln (a^b ) = b \ln (a) \).
\( (x-1) \ln (3) \le \ln (27) \).
4. Solve for \(x\): \( x \ln (3) - \ln (3) \le \ln (27) \).
Add \( \ln (3) \) to both sides: \( x \ln (3) \le \ln (27) + \ln (3) \).
Divide both sides by \( \ln (3) \): \( x \le \frac{\ln (27)+\ln (3)}{\ln (3)} \).
5. Calculate the numerical values: \( x \le \frac{\ln (3^3) + \ln (3)}{ \ln (3)} = 4 \).
So the solution to the exponential inequality \( 3^{x-1} \le 27\) is \( x \le 4 \).

Example 3: \( 5e^{2x} < 100 \)
1. Isolate the exponential term: Divide both sides by 5.
\( e^{2x} < 20 \)
2. Apply the natural logarithm to both sides: \( \ln(e^{2x}) < \ln (20) \).
3. Simplify using the logarithm properties: \( \ln (a^b ) = b \ln (a) \).
\( 2x \ln (e) < \ln (20) \)
Since \( \ln (e) = 1 \), the inequality simplifies to: \( 2x < \ln (20) \)
4. Solve for \(x\): Divide both sides by 2.
\( x < \frac{\ln (20)}{2} \)
5. Calculate the numerical values: \( x < \frac{\ln (20)}{2} \approx 1.4979 \).
So the solution to the exponential inequality \( 5e^{2x} < 100 \) is \( x < \approx 1.4979 \).

Logarithmic inequality☰

A logarithmic inequality is an inequality involving logarithmic functions. Logarithmic functions are the inverse of exponential functions and have the form: \(y= \log_b x \).
Where \(b\) is the base of the logarithm and \(x\) is the argument. In this context, a logarithmic inequality is an inequality that contains a logarithmic function, such as: \( \log_b f(x) \le \log_b g(x) \) or \(\log_b f(x) \ge \log_b g(x) \).

To understand and solve logarithmic inequalities, it is important to know some properties of logarithms:

• Logarithmic identity: \( \log_b\ b^x = x \)
• Change of base formula: \(\log_b\ x =\frac{\log_c x}{\log_c b} \), where \(c\) is another base
• Product rule: \( \log_b x \cdot y = \log_b x + \log_b y \)
• Quotient rule: \( \log_b \frac{x}{y} = \log_b x - \log_b y \)
• Power rule: \( \log_b x^n = n \log_b x \)

Now, let's discuss how to solve a logarithmic inequality. The process generally involves the following steps:
1. Isolate the logarithm on one side of the inequality.
2. Apply properties of logarithms to simplify the inequality if possible.
3. Remove the logarithm from the inequality. This can often be done by exponentiating both sides with the base of the logarithm. Keep in mind that if the base is between 0 and 1, the direction of the inequality will be reversed.
4. Solve the resulting inequality for \(x\).
5. Check your solution to make sure it is valid, as some transformations can introduce extraneous solutions.

Let's illustrate this process with an example:
Solve the inequality \( \log_2 (x^2–6x+8) \ge 1 \).
1. The logarithm is already isolated.
2. We don't need to simplify the inequality.
3. Remove the logarithm by exponentiating both sides with the base of 2: \( 2^{\log_2 (x^2-6x+8)} \ge 2^1 \).
This simplifies to: \( x^2-6x+8 \ge 2 \).
4. Solve the inequality: \( x^2-6x+6 \ge 0 \).
We will find the critical points by applying the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
With \(a = 1 \), \(b = -6\), and \(c = 6\), we get:
\( x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(6)}}{2(1)} = \frac{6 \pm \sqrt{12}}{2} \)

The two critical points are \( x = 3 - \sqrt{3} \) and \( x = 3 + \sqrt{3} \). We can analyze the sign of the inequality between the critical points: For \( x < 3 - \sqrt{3} \), the inequality is positive.
For \( 3 - \sqrt{3} < x < 3 + \sqrt{3} \), the inequality is negative.
For \( x > 3 + \sqrt{3} \), the inequality is positive.
5. Considering the inequality is nonstrict \( (\ge ) \), the solution set is \( x \le 3 - \sqrt{3} \) or \( x \ge 3 + \sqrt{3} \), or in interval notation, \( (-\infty, 3 - \sqrt{3}] \cup [3 + \sqrt{3}, \infty) \).
This is the correct solution to the logarithmic inequality \( \log_2 (x^2–6x+8) \ge 1 \). The general approach outlined here can be used to solve most logarithmic inequalities.