Math Problem Statement

Evaluate log_4(32).

Solution

The problem asks to evaluate log432\log_4 32.

Solution:

The logarithmic equation log432=x\log_4 32 = x can be rewritten as: 4x=324^x = 32

Now express 32 as a power of 4. We know that 32=2532 = 2^5, and 4=224 = 2^2. Thus: 4x=(22)x=22x4^x = (2^2)^x = 2^{2x} and 22x=252^{2x} = 2^5

Equating the exponents: 2x=52x = 5

Solve for xx: x=52x = \frac{5}{2}

Thus: log432=52\log_4 32 = \frac{5}{2}

Final Answer:

log432=52\log_4 32 = \frac{5}{2}

Would you like further explanation on logarithms or properties of exponents?

Here are some related questions for practice:

  1. Evaluate log264\log_2 64.
  2. Simplify log525\log_5 25.
  3. Solve log381=x\log_3 81 = x.
  4. Convert log101000\log_{10} 1000 to exponential form.
  5. What is the value of log816\log_8 16?

Tip: When solving logarithms, always check if the argument (e.g., 32 in this case) can be expressed as a power of the base.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Exponents
Equation Solving

Formulas

log_b(a) = x implies b^x = a
a^(mn) = (a^m)^n

Theorems

Properties of Logarithms
Exponentiation Rules

Suitable Grade Level

Grades 9-12