Problem 11 - College Algebra Essentials

Question

Write each equation in its equivalent logarithmic form. $$2^{-4}=\frac{1}{16}$$

Step-by-Step Solution

Answer

The logarithmic form of the equation \(2^{-4} = \frac{1}{16}\) is \(\log_2 (\frac{1}{16}) = -4\).

Step 1: Understand the exponential form

Given the exponential equation \(2^{-4} = \frac{1}{16}\), the base is 2, the exponent is -4 and the result is \(\frac{1}{16}\)

Step 2: Apply the property of logarithms

By applying the property of logarithms where \(\log_b (a) = n\) is the same as \(b^n = a\), the exponential equation can be converted into a logarithmic equation.

Step 3: Write the equation in logarithmic form

Using the base, result and the exponent from the exponential form, the equation becomes \(\log_2 (\frac{1}{16}) = -4\).