Question
Write the following exponential equation in logarithmic form \( 6^{-2}=\frac{1}{36} \)
Ask by Higgins Newton. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The exponential equation \( 6^{-2} = \frac{1}{36} \) can be written in logarithmic form as \( \log_{6}\left(\frac{1}{36}\right) = -2 \).
Solution
Given the exponential equation
\[
6^{-2}=\frac{1}{36}
\]
we start with the definition of logarithms. In general, the equation
\[
a^x = b
\]
can be rewritten in logarithmic form as
\[
\log_a(b)=x.
\]
Identifying \( a=6 \), \( x=-2 \), and \( b=\frac{1}{36} \), we can write the logarithmic form as
\[
\log_{6}\left(\frac{1}{36}\right)=-2.
\]
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Simplify this solution The Deep Dive
To rewrite the exponential equation \( 6^{-2} = \frac{1}{36} \) in logarithmic form, we recognize that if \( a^b = c \), then \( \log_a(c) = b \). Here, the base is 6, the exponent is -2, and the result is \( \frac{1}{36} \). Therefore, we can express this in logarithmic form as: \[ \log_6\left(\frac{1}{36}\right) = -2 \] This means that 6 raised to the power of -2 gives us \( \frac{1}{36} \).
