Problem 11 - College Algebra

Question

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

Step-by-Step Solution

Answer

The logarithmic form of the given equation is \( \log_{2}\left(\frac{1}{16}\right) = -4 \).

Step 1: Identify the Base

Start by identifying the base in the exponential equation. The base is represented by the number being multiplied by itself, which in this case is \(2\). This is the base of the logarithmic function.

Step 2: Identify the Exponent

The next step is to identify the exponent in the exponential equation. The exponent represents the number of times the base is multiplied by itself and is represented as \(-4\) in the equation. This will become the argument of the logarithm.

Step 3: Identify the Result

Finally, the result of the exponential equation, in this case, \(\frac{1}{16}\), will be the characteristic of the logarithm.

Step 4: Convert to Logarithmic Form

The logarithmic form of the equation is: \[ \log_{base}({result}) = exponent \]. Substituting the identified elements into this form, the equation becomes: \[ \log_{2}\left(\frac{1}{16}\right) = -4 \].