Question

exponential and logarithmic functions
converting between logarithmic and exponential equations
rewrite each equation as requested.
(a) rewrite as a logarithmic equation.
$6^{-2} = \frac{1}{36}$
$\log_{\square}\square = \square$
(b) rewrite as an exponential equation.
$\log_{4} 64 = 3$
$\square^{\square} = \square$

Explanation:

Part (a)

Step 1: Recall the logarithmic form

The exponential equation \(a^b = c\) can be rewritten in logarithmic form as \(\log_a c = b\).

Step 2: Apply the formula to \(6^{-2}=\frac{1}{36}\)

Here, \(a = 6\), \(b=-2\), and \(c=\frac{1}{36}\). So the logarithmic equation is \(\log_{6}\frac{1}{36}=-2\).

Part (b)

Step 1: Recall the exponential form

The logarithmic equation \(\log_a c = b\) can be rewritten in exponential form as \(a^b = c\).

Step 2: Apply the formula to \(\log_4 64 = 3\)

Here, \(a = 4\), \(b = 3\), and \(c = 64\). So the exponential equation is \(4^3=64\).

Answer:

(a) \(\log_{6}\frac{1}{36}=-2\) (So the boxes are filled as base \(6\), argument \(\frac{1}{36}\), and result \(-2\))
(b) \(4^3 = 64\) (So the boxes are filled as base \(4\), exponent \(3\), and result \(64\))