Solving Logarithmic Equations

Overview

Logarithmic equations are equations where the variable appears inside a logarithm. For example:

\( \log(x) = 2 \)

To solve logarithmic equations, we use one or more of the following strategies:

  • Rewriting in exponential form: Use the definition of a logarithm to rewrite the equation as an exponential equation.
  • Using logarithmic properties: Simplify the equation by applying properties of logarithms such as the product, quotient, or power rules.
  • Isolating the logarithm: When there is only one logarithm, isolate it before solving.

Key Techniques

1. Rewriting in Exponential Form

Recall that \( \log_b(a) = c \) is equivalent to \( b^c = a \).

Example: Solve \( \log_2(x) = 5 \)

Step 1: Rewrite as \( 2^5 = x \)

Step 2: Simplify: \( x = 32 \)

2. Using Logarithmic Properties

  • Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
  • Quotient Rule: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)
  • Power Rule: \( \log_b(m^n) = n \cdot \log_b(m) \)

Example: Solve \( \log_3(x) + \log_3(2) = 4 \)

Step 1: Use the product rule: \( \log_3(2x) = 4 \)

Step 2: Rewrite in exponential form: \( 3^4 = 2x \)

Step 3: Solve for \( x \): \( x = \frac{81}{2} \)

3. Solving with Multiple Logarithms

Combine logarithms on one side before solving:

Example: Solve \( \log_5(x) - \log_5(2) = 1 \)

Step 1: Use the quotient rule: \( \log_5\left(\frac{x}{2}\right) = 1 \)

Step 2: Rewrite in exponential form: \( 5^1 = \frac{x}{2} \)

Step 3: Solve for \( x \): \( x = 10 \)

Practice Questions

  1. Question 1: Solve \( \log_4(x) = 3 \).
    Solution

    Step 1: Rewrite in exponential form: \( 4^3 = x \)

    Step 2: Simplify: \( x = 64 \)

  2. Question 2: Solve \( \log_2(x) + \log_2(8) = 5 \).
    Solution

    Step 1: Use the product rule: \( \log_2(8x) = 5 \)

    Step 2: Rewrite in exponential form: \( 2^5 = 8x \)

    Step 3: Solve for \( x \): \( x = 4 \)

  3. Question 3: Solve \( \log(x) = -1 \) (base 10).
    Solution

    Step 1: Rewrite in exponential form: \( 10^{-1} = x \)

    Step 2: Simplify: \( x = 0.1 \)

  4. Question 4: Solve \( \log_3(x - 2) = 2 \).
    Solution

    Step 1: Rewrite in exponential form: \( 3^2 = x - 2 \)

    Step 2: Simplify: \( 9 = x - 2 \)

    Step 3: Solve for \( x \): \( x = 11 \)

  5. Question 5: Solve \( \log_7(x + 3) - \log_7(2) = 1 \).
    Solution

    Step 1: Use the quotient rule: \( \log_7\left(\frac{x + 3}{2}\right) = 1 \)

    Step 2: Rewrite in exponential form: \( 7^1 = \frac{x + 3}{2} \)

    Step 3: Solve for \( x \): \( x = 11 \)

Extra Practice

  • Solve \( \log_5(3x) = 2 \)
  • Solve \( \log_6(x) + \log_6(4) = 3 \)
  • Solve \( \log(x - 1) = 0 \) (base 10)
  • Solve \( \log_2(x^2) = 6 \)
  • Solve \( \ln(x) = 2 \)
→Exponential and Logarithmic Models