Question 2:

For any positive number [tex]\( b \)[/tex] not equal to 1 and any number or variable [tex]\( n \)[/tex], evaluate the expression:

[tex]\[ b^{\log_8 n} = \][/tex]

A. [tex]\( b \)[/tex]
B. [tex]\( n \)[/tex]
C. [tex]\( \log n \)[/tex]
D. [tex]\( \log b \)[/tex]



Answer :

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]