Exponential and Logarithmic Inequalities

Overview

Exponential and logarithmic inequalities are solved by applying the properties of exponents and logarithms while carefully considering the inequality's direction. These inequalities often require transformations and the application of logarithmic or exponential rules.

Key Concepts

Exponential Inequalities

To solve inequalities involving exponential functions, use logarithms to "bring down" the exponent. Consider the base of the exponential function:

  • If the base \( b > 1 \), the inequality's direction remains unchanged.
  • If the base \( 0 < b < 1 \), the inequality's direction is reversed.

Example:

  • Solve \( 3^{x} > 9 \): Rewrite as \( x > \log_3(9) \), so \( x > 2 \).

Logarithmic Inequalities

To solve inequalities involving logarithmic functions, rewrite the logarithmic equation in its exponential form. Make sure the argument of the logarithm is positive:

  • \( \log_b(x) > c \): Rewrite as \( x > b^c \).
  • \( \log_b(x) < c \): Rewrite as \( x < b^c \).

Example:

  • Solve \( \log_2(x) \leq 3 \): Rewrite as \( x \leq 2^3 \), so \( x \leq 8 \).

Steps for Solving

  1. Isolate the exponential or logarithmic term.
  2. Apply logarithmic or exponential rules to rewrite the inequality.
  3. Solve for the variable.
  4. Check for any restrictions (e.g., arguments of logarithms must be positive).

Practice Questions

  1. Question 1: Solve \( 2^x \leq 16 \).
    Solution

    \( x \leq \log_2(16) = 4 \)

  2. Question 2: Solve \( 3^{x+1} > 27 \).
    Solution

    Rewrite as \( x+1 > \log_3(27) = 3 \). Thus, \( x > 2 \).

  3. Question 3: Solve \( \log(x) > 2 \).
    Solution

    Rewrite as \( x > 10^2 = 100 \).

  4. Question 4: Solve \( \ln(x-1) \leq 3 \).
    Solution

    Rewrite as \( x-1 \leq e^3 \). Thus, \( x \leq e^3 + 1 \).

  5. Question 5: Solve \( \log_5(3x) \geq 1 \).
    Solution

    Rewrite as \( 3x \geq 5^1 = 5 \). Divide by 3: \( x \geq \frac{5}{3} \).

Extra Practice

  • Solve \( 4^{2x-1} \geq 64 \).
  • Solve \( \ln(x+4) > 2 \).
  • Solve \( \log_7(x-3) < 0 \).
  • Solve \( 5^{x} - 3 < 22 \).
  • Solve \( \log_2(2x+5) \geq 3 \).