Certainly! Let's find the exact value of the expression [tex]\( 8^{\log_8 5} \)[/tex] by using properties of logarithms.
1. Understanding the expression:
The expression [tex]\( 8^{\log_8 5} \)[/tex] consists of a base, which is 8, raised to a logarithm with the same base, [tex]\( \log_8 5 \)[/tex].
2. Applying the property of logarithms:
There is a fundamental property of logarithms that states [tex]\( a^{\log_a b} = b \)[/tex]. This means that if you have a number [tex]\( a \)[/tex] raised to the power of the logarithm of another number [tex]\( b \)[/tex] with the same base [tex]\( a \)[/tex], the result simplifies directly to [tex]\( b \)[/tex].
3. Using the property:
In our expression, [tex]\( a = 8 \)[/tex] and [tex]\( b = 5 \)[/tex]. Therefore, we can apply the property as follows:
[tex]\[
8^{\log_8 5} = 5
\][/tex]
Thus, the exact value of the expression [tex]\( 8^{\log_8 5} \)[/tex] is [tex]\( 5 \)[/tex].
