Question

rewrite each equation as requested.
(a) rewrite as an exponential equation.
\\(\log_{8} 1 = 0\\)
\\(\square^{\square} = \square\\)
(b) rewrite as a logarithmic equation.
\\(3^{4} = 81\\)
\\(\log_{\square} \square = \square\\)

Explanation:

Part (a)

Step 1: Recall the logarithmic to exponential conversion formula

The definition of a logarithm states that if $\log_b a = c$, then the equivalent exponential form is $b^c = a$.

Step 2: Identify the values of \(b\), \(a\), and \(c\) from \(\log_8 1 = 0\)

Here, the base \(b = 8\), the argument \(a = 1\), and the result \(c = 0\).

Step 3: Apply the conversion formula

Substituting these values into the exponential form \(b^c = a\), we get \(8^0 = 1\).

Part (b)

Step 1: Recall the exponential to logarithmic conversion formula

If \(b^c = a\), then the equivalent logarithmic form is \(\log_b a = c\).

Step 2: Identify the values of \(b\), \(c\), and \(a\) from \(3^4 = 81\)

Here, the base \(b = 3\), the exponent \(c = 4\), and the result \(a = 81\).

Step 3: Apply the conversion formula

Substituting these values into the logarithmic form \(\log_b a = c\), we get \(\log_3 81 = 4\).

Answer:

(a) \(8^0 = 1\)
(b) \(\log_3 81 = 4\)