Solving Exponential Equations

Overview

Exponential equations are equations in which the unknown variable appears in the exponent. For example:

\( 2^x = 16 \)

To solve exponential equations, we use one or more of the following techniques:

  • Rewriting both sides with the same base: If the bases are the same, set the exponents equal to each other.
  • Using logarithms: If the bases cannot be made the same, take the logarithm of both sides to bring the exponent down.

Key Techniques

1. Rewriting with the Same Base

If both sides of the equation can be written with the same base, set the exponents equal and solve:

Example: Solve \( 2^x = 16 \) Step 1: Rewrite 16 as \( 2^4 \): \( 2^x = 2^4 \) Step 2: Set exponents equal: \( x = 4 \)

2. Using Logarithms

If the bases are not easily rewritten, use logarithms to solve:

Example: Solve \( 3^x = 7 \) Step 1: Take the natural log (ln) of both sides: \( \ln(3^x) = \ln(7) \) Step 2: Use the property of logarithms: \( x \ln(3) = \ln(7) \) Step 3: Solve for \( x \): \( x = \frac{\ln(7)}{\ln(3)} \)

Practice Questions

  1. Question 1: Solve \( 5^x = 125 \).
    Solution

    Step 1: Rewrite 125 as \( 5^3 \): \( 5^x = 5^3 \) Step 2: Set exponents equal: \( x = 3 \)

  2. Question 2: Solve \( 2^{3x} = 32 \).
    Solution

    Step 1: Rewrite 32 as \( 2^5 \): \( 2^{3x} = 2^5 \) Step 2: Set exponents equal: \( 3x = 5 \) Step 3: Solve for \( x \): \( x = \frac{5}{3} \)

  3. Question 3: Solve \( 10^x = 1000 \).
    Solution

    Step 1: Rewrite 1000 as \( 10^3 \): \( 10^x = 10^3 \) Step 2: Set exponents equal: \( x = 3 \)

  4. Question 4: Solve \( 4^x = 10 \).
    Solution

    Step 1: Take \( \ln \) of both sides: \( \ln(4^x) = \ln(10) \) Step 2: Use logarithm rule: \( x \ln(4) = \ln(10) \) Step 3: Solve for \( x \): \( x = \frac{\ln(10)}{\ln(4)} \)

  5. Question 5: Solve \( 7^{2x - 1} = 49 \).
    Solution

    Step 1: Rewrite 49 as \( 7^2 \): \( 7^{2x - 1} = 7^2 \) Step 2: Set exponents equal: \( 2x - 1 = 2 \) Step 3: Solve for \( x \): \( 2x = 3 \) \( x = \frac{3}{2} \)

  6. Question 6: Solve \( e^x = 20 \) (where \( e \) is the natural base).
    Solution

    Step 1: Take the natural log (ln) of both sides: \( \ln(e^x) = \ln(20) \) Step 2: Use logarithm rule: \( x = \ln(20) \)

Extra Practice

  • Solve \( 9^{x+2} = 81 \)
  • Solve \( 2^x = 10 \)
  • Solve \( 6^{2x} = 36 \)
  • Solve \( e^{3x} = 50 \)
→Solving Logarithmic Equations