To rewrite the exponential equation \(2^{-4} = \frac{1}{16}\) as a logarithmic equation, follow these steps:
1. Understand the Exponential Form:
The given equation \(2^{-4} = \frac{1}{16}\) is in the form of \(a^b = c\), where:
- \(a\) (the base) is 2,
- \(b\) (the exponent) is -4,
- \(c\) (the result) is \(\frac{1}{16}\).
2. Identify Logarithmic Form:
The logarithmic form corresponding to the exponential equation \(a^b = c\) is \(\log_a(c) = b\).
3. Substitute the Values:
- The base (\(a\)) is 2,
- The result (\(c\)) is \(\frac{1}{16}\),
- The exponent (\(b\)) is -4.
So, substituting these into the logarithmic form \(\log_a(c) = b\) gives us:
[tex]\[
\log_2\left(\frac{1}{16}\right) = -4
\][/tex]
Thus, the exponential equation \(2^{-4} = \frac{1}{16}\) when rewritten as a logarithmic equation is:
[tex]\[
\log_2\left(0.0625\right) = -4
\][/tex]
Note: \(\frac{1}{16}\) can be expressed as the decimal \(0.0625\).
Therefore, the logarithmic equation is:
[tex]\[
\log_2(0.0625) = -4
\][/tex]
