The Change of Base Formula

Oct 13, 2025 by miodeya.com 117 views

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them is crucial in various fields such as physics, engineering, and computer science. In this article, we will focus on solving the exponential equation $$3^{x-8}=8$$ using the change of base formula $$\log _b y=\frac{\log y}{\log b}$$. This formula allows us to change the base of a logarithm, making it easier to solve exponential equations.

The Change of Base Formula

The change of base formula is a fundamental concept in mathematics that allows us to change the base of a logarithm. It is given by the equation $$\log _b y=\frac{\log y}{\log b}$$. This formula can be used to change the base of a logarithm from one base to another.

Applying the Change of Base Formula

To solve the exponential equation $$3^{x-8}=8$$, we can use the change of base formula. We can start by taking the logarithm of both sides of the equation. This will give us $$\log 3^{x-8}=\log 8$$. Using the change of base formula, we can rewrite the left-hand side of the equation as $$\frac{\log 3^{x-8}}{\log 3}=\frac{\log 8}{\log 3}$$. Simplifying the left-hand side, we get $$(x-8)\log 3=\log 8$$.

Solving for x

Now that we have the equation $$(x-8)\log 3=\log 8$$, we can solve for x. We can start by dividing both sides of the equation by $$\log 3$$. This will give us $$(x-8)=\frac{\log 8}{\log 3}$$. Now, we can add 8 to both sides of the equation to get $$x=8+\frac{\log 8}{\log 3}$$.

Evaluating the Expression

To evaluate the expression $$x=8+\frac{\log 8}{\log 3}$$, we can use a calculator. Plugging in the values, we get $$x=8+\frac{0.9031}{0.4771}$$.

Simplifying the Expression

Simplifying the expression $$x=8+\frac{0.9031}{0.4771}$$, we get $$x=8+1.894$$. Evaluating the expression, we get $$x=9.894$$.

Conclusion

In this article, we used the change of base formula to solve the exponential equation $$3^{x-8}=8$$. We started by taking the logarithm of both sides of the equation and then used the change of base formula to rewrite the left-hand side. We then solved for x and evaluated the expression using a calculator. The final answer is $$x=9.894$$.

Answer

The final answer is $$\boxed{9.893}$$.

Discussion

The change of base formula is a powerful tool for solving exponential equations. It allows us to change the base of a logarithm, making it easier to solve equations. In this article, we used the change of base formula to solve the exponential equation $$3^{x-8}=8$$. We started by taking the logarithm of both sides of the equation and then used the change of base formula to rewrite the left-hand side. We then solved for x and evaluated the expression using a calculator. The final answer is $$x=9.894$$.

References

[1] "Change of Base Formula" by Math Open Reference. [Online]. Available: https://www.mathopenref.com/logarithmchangeofbase.html
[2] "Exponential Equations" by Math Is Fun. [Online]. Available: https://www.mathisfun.com/algebra/exponential-equations.html

Keywords

Exponential equations
Change of base formula
Logarithms
Mathematics
Algebra
Solving Exponential Equations Using the Change of Base Formula: Q&A ====================================================================

Introduction

In our previous article, we used the change of base formula to solve the exponential equation $$3^{x-8}=8$$. We received many questions from readers who wanted to know more about the change of base formula and how to apply it to solve exponential equations. In this article, we will answer some of the most frequently asked questions about the change of base formula and solving exponential equations.

Q: What is the change of base formula?

A: The change of base formula is a mathematical formula that allows us to change the base of a logarithm. It is given by the equation $$\log _b y=\frac{\log y}{\log b}$$. This formula can be used to change the base of a logarithm from one base to another.

Q: How do I apply the change of base formula to solve exponential equations?

A: To apply the change of base formula to solve exponential equations, you need to follow these steps:

Take the logarithm of both sides of the equation.
Use the change of base formula to rewrite the left-hand side of the equation.
Simplify the equation and solve for x.

Q: What are some common mistakes to avoid when using the change of base formula?

A: Here are some common mistakes to avoid when using the change of base formula:

Not using the correct base for the logarithm.
Not simplifying the equation correctly.
Not solving for x correctly.

Q: Can I use the change of base formula to solve logarithmic equations?

A: Yes, you can use the change of base formula to solve logarithmic equations. The change of base formula can be used to change the base of a logarithm, making it easier to solve logarithmic equations.

Q: What are some real-world applications of the change of base formula?

A: The change of base formula has many real-world applications, including:

Calculating the pH of a solution.
Determining the concentration of a solution.
Solving problems in physics and engineering.

Q: Can I use a calculator to solve exponential equations using the change of base formula?

A: Yes, you can use a calculator to solve exponential equations using the change of base formula. In fact, using a calculator can make it easier to solve exponential equations and reduce the risk of errors.

Q: What are some tips for solving exponential equations using the change of base formula?

A: Here are some tips for solving exponential equations using the change of base formula:

Make sure to use the correct base for the logarithm.
Simplify the equation correctly.
Solve for x correctly.

Conclusion

In this article, we answered some of the most frequently asked questions about the change of base formula and solving exponential equations. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the change of base formula. If you have any further questions, please don't hesitate to ask.

References

[1] "Change of Base Formula" by Math Open Reference. [Online]. Available: https://www.mathopenref.com/logarithmchangeofbase.html
[2] "Exponential Equations" by Math Is Fun. [Online]. Available: https://www.mathisfun.com/algebra/exponential-equations.html

Keywords

Exponential equations
Change of base formula
Logarithms
Mathematics
Algebra
Calculus