Question

For any positive number b not equal to 1 and any number or variable n, evaluate the following expression.

Answer 1

Answer:

 $log_{b} b^{n}$ = n.

Step-by-step explanation:

Given :  $log_{b} b^{n}$ , b is any positive number not equal to 1 and n is any number.

To find : Evaluate the expression  $log_{b} b^{n}$.

Formula used : $log_{b} x^{y}$. = y ∙ $log_{b}$(x) and

$log_{b}(b) = 1.Solution : We have [tex]log_{b} b^{n}$.

By logarithm rule :  $log_{b} x^{y}$. = y ∙ $log_{b}$(x).

Then  $log_{b} b^{n}$ = n∙ $log_{b}$(b).

By logarithm rule :  $log_{b}(b) = 1.Now, n∙ [tex]log_{b}$(b) = n.

Therefore,  $log_{b} b^{n}$ = n.

Answer 2

The answer is n.

If: $log_x(y^{z}) = z*log_x(y)$
Then: $log_b( b^{n}) = n*log_b(b)$

If: $log_x(y) = \frac{log_z(y)}{log_z(x)}$
Then: $log_b(b) = \frac{log_z(b)}{log_z(b)} =1$

Therefore:
$log_b( b^{n}) = n*log_b(b) =n*1=n$