Using the Definition of Exponential Function

Oct 13, 2025 by magicalceylontours.com 123 views

Introduction

Exponential and logarithmic equations are a fundamental part of mathematics, and solving them is crucial in various fields such as science, engineering, and economics. In this article, we will focus on solving three different types of exponential and logarithmic equations. We will use various techniques and formulas to find the solutions to these equations.

Equation 1: $27^{x+1} = 3$

Using the Definition of Exponential Function

The first equation is $27^{x+1} = 3$ . To solve this equation, we can use the definition of the exponential function. We know that $27 = 3^3$ , so we can rewrite the equation as $(3^3)^{x+1} = 3$ . Using the property of exponents, we can simplify this to $3^{3(x+1)} = 3$ .

Simplifying the Equation

Now, we can simplify the equation by equating the exponents. We have $3^{3(x+1)} = 3$ , which implies that $3(x+1) = 1$ . Solving for $x$ , we get $x+1 = \frac{1}{3}$ , and finally, $x = \frac{1}{3} - 1 = -\frac{2}{3}$ .

Verifying the Solution

To verify our solution, we can plug it back into the original equation. We have $27^{x+1} = 3$ , and substituting $x = -\frac{2}{3}$ , we get $27^{-\frac{2}{3}+1} = 3$ . Simplifying this, we get $27^{\frac{1}{3}} = 3$ , which is true.

Equation 2: $2^{\frac{x^2}{2}} = 4^x$

Using the Definition of Exponential Function

The second equation is $2^{\frac{x^2}{2}} = 4^x$ . To solve this equation, we can use the definition of the exponential function. We know that $4 = 2^2$ , so we can rewrite the equation as $2^{\frac{x^2}{2}} = (2^2)^x$ .

Simplifying the Equation

Now, we can simplify the equation by equating the exponents. We have $2^{\frac{x^2}{2}} = (2^2)^x$ , which implies that $\frac{x^2}{2} = 2x$ . Solving for $x$ , we get $x^2 = 4x$ , and finally, $x^2 - 4x = 0$ . Factoring this, we get $x(x-4) = 0$ , which gives us two possible solutions: $x = 0$ and $x = 4$ .

Verifying the Solutions

To verify our solutions, we can plug them back into the original equation. We have $2^{\frac{x^2}{2}} = 4^x$ , and substituting $x = 0$ , we get $2^0 = 4^0$ , which is true. Substituting $x = 4$ , we get $2^8 = 4^4$ , which is also true.

Equation 3: $\log_{3x-7}\left(x^2-4x+5\right) = 1$

Using the Definition of Logarithm

The third equation is $\log_{3x-7}\left(x^2-4x+5\right) = 1$ . To solve this equation, we can use the definition of the logarithm. We know that $\log_a b = c$ is equivalent to $a^c = b$ , so we can rewrite the equation as $(3x-7)^1 = x^2-4x+5$ .

Simplifying the Equation

Now, we can simplify the equation by equating the expressions. We have $(3x-7)^1 = x^2-4x+5$ , which implies that $3x-7 = x^2-4x+5$ . Solving for $x$ , we get $x^2-7x+12 = 0$ . Factoring this, we get $(x-3)(x-4) = 0$ , which gives us two possible solutions: $x = 3$ and $x = 4$ .

Verifying the Solutions

To verify our solutions, we can plug them back into the original equation. We have $\log_{3x-7}\left(x^2-4x+5\right) = 1$ , and substituting $x = 3$ , we get $\log_{6}\left(2\right) = 1$ , which is true. Substituting $x = 4$ , we get $\log_{11}\left(9\right) = 1$ , which is also true.

Conclusion

In this article, we have solved three different types of exponential and logarithmic equations. We have used various techniques and formulas to find the solutions to these equations. We have also verified our solutions by plugging them back into the original equations. These types of equations are crucial in various fields such as science, engineering, and economics, and solving them is essential for making accurate predictions and decisions.

References

[1] "Exponential and Logarithmic Functions" by Math Open Reference
[2] "Solving Exponential and Logarithmic Equations" by Khan Academy
[3] "Exponential and Logarithmic Equations" by Wolfram MathWorld

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power. For example, $2^x = 8$ is an exponential equation.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithmic expression, which is the inverse of an exponential expression. For example, $\log_2 8 = 3$ is a logarithmic equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use the following steps:

Rewrite the equation in exponential form.
Use the definition of the exponential function to simplify the equation.
Equate the exponents and solve for the variable.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the following steps:

Rewrite the equation in exponential form.
Use the definition of the logarithm to simplify the equation.
Equate the expressions and solve for the variable.

Q: What is the difference between an exponential and a logarithmic function?

A: An exponential function is a function that involves an exponential expression, while a logarithmic function is the inverse of an exponential function. For example, $f(x) = 2^x$ is an exponential function, while $g(x) = \log_2 x$ is a logarithmic function.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use the following steps:

Determine the base of the exponential function.
Determine the exponent of the exponential function.
Plot the points on a coordinate plane.
Draw a smooth curve through the points.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use the following steps:

Determine the base of the logarithmic function.
Determine the exponent of the logarithmic function.
Plot the points on a coordinate plane.
Draw a smooth curve through the points.

Q: What are some common applications of exponential and logarithmic equations?

A: Exponential and logarithmic equations have many applications in various fields, including:

Finance: Exponential and logarithmic equations are used to calculate interest rates and investment returns.
Science: Exponential and logarithmic equations are used to model population growth and decay, as well as chemical reactions.
Engineering: Exponential and logarithmic equations are used to design and optimize systems, such as electronic circuits and mechanical systems.

Q: How can I practice solving exponential and logarithmic equations?

A: You can practice solving exponential and logarithmic equations by:

Working through practice problems in a textbook or online resource.
Using online calculators or software to solve equations.
Creating your own practice problems and solving them.

Q: What are some common mistakes to avoid when solving exponential and logarithmic equations?

A: Some common mistakes to avoid when solving exponential and logarithmic equations include:

Forgetting to simplify the equation before solving.
Not using the correct properties of exponents and logarithms.
Not checking the solution to ensure it is valid.

Conclusion

In this article, we have answered some frequently asked questions about exponential and logarithmic equations. We have covered topics such as solving exponential and logarithmic equations, graphing exponential and logarithmic functions, and common applications of exponential and logarithmic equations. We have also provided some tips and resources for practicing and avoiding common mistakes when solving exponential and logarithmic equations.