Question

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

logb(b^n) =
with logb being the base

Answer 1

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) = (log_z(y))/(log_z(x))`
Then:
`log_b(b) = (log_z(b))/(log_z(b)) =1`

Therefore:

`log_b( b^(n)) = n*log_b(b) =n*1=n`

Answer 2

`log_(b) b^(n)` = n.

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.</p><p>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.</p><p>Now, n∙ [tex]log_(b)`(b) = n.

Therefore,
`log_(b) b^(n)` = n.