To evaluate the expression [tex]\( b^{\log_8 n} \)[/tex], we'll go through the following steps:
1. Understanding the logarithmic expression:
The expression given is [tex]\( b^{\log_8 n} \)[/tex]. It involves a logarithm with a base [tex]\( 8 \)[/tex].
2. Using the properties of logarithms:
Recall the change of base formula for logarithms. For any positive number [tex]\( a \)[/tex] and any number [tex]\( x \)[/tex]:
[tex]\[
\log_a x = \frac{\log_b x}{\log_b a}
\][/tex]
Here, we can apply the change of base formula to [tex]\( \log_8 n \)[/tex]:
[tex]\[
\log_8 n = \frac{\log n}{\log 8}
\][/tex]
3. Substitute and simplify the exponent:
Now substitute [tex]\( \log_8 n \)[/tex] in the expression [tex]\( b^{\log_8 n} \)[/tex]:
[tex]\[
b^{\log_8 n} = b^{\frac{\log n}{\log 8}}
\][/tex]
4. Using properties of exponents:
Recall that [tex]\( b^{a \cdot b} = (b^a)^b \)[/tex]:
[tex]\[
b^{\frac{\log n}{\log 8}} = \left(b^{\log n}\right)^{\frac{1}{\log 8}}
\][/tex]
So it simplifies to:
[tex]\[
(b^{\log n})^{\frac{1}{\log 8}}
\][/tex]
5. Simplifying using the identity [tex]\( b^{\log_b x} = x \)[/tex]:
Given [tex]\( a^{\log_a x} = x \)[/tex], this identity simplifies [tex]\( b^{\log n} \)[/tex] directly, but it is independent of the base change:
[tex]\[
b^{\log_n} = n
\][/tex]
Therefore, the final simplified result of the expression [tex]\( b^{\log_8 n} \)[/tex] is indeed [tex]\( n \)[/tex].
So, the correct answer is:
[tex]\[
\boxed{n}
\][/tex]
Option B. [tex]\( n \)[/tex]
