Sure, let's work through each part of the problem step-by-step.
### (a) Rewrite \( 6^2 = 36 \) as a logarithmic equation.
We are given the exponential equation:
[tex]\[ 6^2 = 36 \][/tex]
To rewrite this as a logarithmic equation, we use the definition of a logarithm. Specifically, if \( a^b = c \), then this can be rewritten in logarithmic form as \( \log_a(c) = b \).
Here, \( a = 6 \), \( b = 2 \), and \( c = 36 \). Using the definition:
[tex]\[ \log_6(36) = 2 \][/tex]
So the logarithmic equation for \( 6^2 = 36 \) is:
[tex]\[ \log_6(36) = 2 \][/tex]
### (b) Rewrite \( \log_4 \left( \frac{1}{64} \right) = -3 \) as an exponential equation.
We are given the logarithmic equation:
[tex]\[ \log_4 \left( \frac{1}{64} \right) = -3 \][/tex]
To rewrite this as an exponential equation, we use the inverse relationship between exponentiation and logarithms. Specifically, if \( \log_a(b) = c \), then this can be rewritten in exponential form as \( a^c = b \).
Here, \( a = 4 \), \( c = -3 \), and \( b = \frac{1}{64} \). Using the definition:
[tex]\[ 4^{-3} = \frac{1}{64} \][/tex]
So the exponential equation for \( \log_4 \left( \frac{1}{64} \right) = -3 \) is:
[tex]\[ 4^{-3} = \frac{1}{64} \][/tex]
Combining the results, we have:
### Final Answer
(a) \( \log_6(36) = 2 \)
(b) [tex]\( 4^{-3} = \frac{1}{64} \)[/tex]
