Solving Exponential Equations
Overview
When the unknown sits in the exponent, ordinary algebra cannot reach it. Logarithms are the tool that brings it down to ground level. Taking the logarithm of both sides of \(a^x=b\) lets you slide the power out in front, turning an exponential equation into a linear one you can solve at once.
In this lesson you will solve equations like \(2^x=10\) by taking logarithms, using either base 10 or the natural logarithm. The key formula is \(x=\dfrac{\log b}{\log a}\), and you will see how it follows directly from the power law.
Objectives
- Solve equations of the form a to the power x equals b.
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Lesson Note
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Lesson Evaluation
Congratulations on completing the lesson on Solving Exponential Equations. Now that youve explored the key concepts and ideas, its time to put your knowledge to the test. This section offers a variety of practice questions designed to reinforce your understanding and help you gauge your grasp of the material.
You will encounter a mix of question types, including multiple-choice questions, short answer questions, and essay questions. Each question is thoughtfully crafted to assess different aspects of your knowledge and critical thinking skills.
Use this evaluation section as an opportunity to reinforce your understanding of the topic and to identify any areas where you may need additional study. Don't be discouraged by any challenges you encounter; instead, view them as opportunities for growth and improvement.
- To solve a^x = b, the first step is to: A. divide both sides by a B. take logarithms of both sides C. square both sides D. subtract b Answer: B
- Solving 2^x = 10 gives x equal to: A. log 2 / log 10 B. log 10 / log 2 C. log(10/2) D. 10/2 Answer: B
- To 2 decimal places, the solution of 2^x = 10 is: A. 2.32 B. 3.32 C. 5.00 D. 1.43 Answer: B
- Solve e^x = 7 (to 2 decimal places). A. 0.85 B. 1.95 C. 2.71 D. 7.00 Answer: B
- The step x log a = log b leads to x equal to: A. log b - log a B. log b / log a C. log(b - a) D. log a / log b Answer: B
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