Sure, let's rewrite each given equation step-by-step as requested.
### (a) Rewrite as an exponential equation.
The given logarithmic equation is:
[tex]\[ \log_3(81) = 4 \][/tex]
To rewrite it as an exponential equation, recall that [tex]\(\log_b(a) = c\)[/tex] means [tex]\(b^c = a\)[/tex]. Here, [tex]\(b = 3\)[/tex], [tex]\(a = 81\)[/tex], and [tex]\(c = 4\)[/tex]. Therefore:
[tex]\[ 3^4 = 81 \][/tex]
So, the exponential equation is:
[tex]\[ 3^4 = 81 \][/tex]
### (b) Rewrite as a logarithmic equation.
The given exponential equation is:
[tex]\[ 8^{-1} = \frac{1}{8} \][/tex]
To rewrite it as a logarithmic equation, recall that if [tex]\(b^c = a\)[/tex], then [tex]\(\log_b(a) = c\)[/tex]. Here, [tex]\(b = 8\)[/tex], [tex]\(a = \frac{1}{8}\)[/tex], and [tex]\(c = -1\)[/tex]. Therefore:
[tex]\[ \log_8 \left( \frac{1}{8} \right) = -1 \][/tex]
So, the logarithmic equation is:
[tex]\[ \log_8\left( \frac{1}{8} \right) = -1 \][/tex]
Putting it all together:
(a) [tex]\( 3^4 = 81 \)[/tex]
(b) [tex]\(\log_8 \left( \frac{1}{8} \right) = -1\)[/tex]
