Evaluate log base c of 16* log base 2 of c

$\log_{c} (16) \cdot \log_{2} (c)$

Step 1

Rewrite $\log_{c} (16)$ using the change of base formula.

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{2} (c)$

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (16)}{\log (c)} \log_{2} (c)$

Step 2

Rewrite $\log_{2} (c)$ using the change of base formula.

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\log (16)}{\log (c)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (16)}{\log (c)} \cdot \frac{\log (c)}{\log (2)}$

Step 3

Cancel the common factor of $\log (c)$ .

Cancel the common factor.

$\frac{\log (16)}{\log (c)} \cdot \frac{\log (c)}{\log (2)}$

Rewrite the expression.

$\log (16) \frac{1}{\log (2)}$

Step 4

Combine $\log (16)$ and $\frac{1}{\log (2)}$ .

$\frac{\log (16)}{\log (2)}$

Step 5

Rewrite $\log (16)$ as $\log (2^{4})$ .

$\frac{\log (2^{4})}{\log (2)}$

Step 6

Expand $\log (2^{4})$ by moving $4$ outside the logarithm.

$\frac{4 \log (2)}{\log (2)}$

Step 7

Cancel the common factor of $\log (2)$ .

Cancel the common factor.

$\frac{4 \log (2)}{\log (2)}$

Divide $4$ by $1$ .

$4$