Math Problem Statement

What is the value of
log 2 1 /32 ?

Solution

To find the value of log2(132)\log_2 \left( \frac{1}{32} \right), we can break it down step by step.

Step 1: Express 32 as a power of 2

We know that: 32=2532 = 2^5 So, we can rewrite the expression as: log2(132)=log2(125)\log_2 \left( \frac{1}{32} \right) = \log_2 \left( \frac{1}{2^5} \right)

Step 2: Apply the logarithmic property for division

Using the property of logarithms logb(1x)=logb(x)\log_b \left( \frac{1}{x} \right) = -\log_b(x), we get: log2(125)=log2(25)\log_2 \left( \frac{1}{2^5} \right) = -\log_2 \left( 2^5 \right)

Step 3: Simplify using the logarithmic identity logb(bx)=x\log_b (b^x) = x

Since log2(25)=5\log_2 (2^5) = 5, the expression becomes: log2(25)=5- \log_2 (2^5) = -5

Thus, the value of log2(132)\log_2 \left( \frac{1}{32} \right) is -5.


Would you like more details on logarithmic properties, or have any other questions?

Related questions:

  1. How do you calculate log381\log_3 81?
  2. What is the change of base formula for logarithms?
  3. Can you simplify log264\log_2 64?
  4. How does the power rule for logarithms work?
  5. What is the inverse of a logarithmic function?

Tip:

To calculate logarithms easily, remember the basic powers of 2, 3, and other common bases.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Exponents
Logarithmic Properties

Formulas

log_b(1/x) = -log_b(x)
log_b(b^x) = x

Theorems

Logarithmic properties
Logarithmic identities

Suitable Grade Level

Grades 7-9