Certainly! To find the value of the logarithmic expression [tex]\(\log_4 32 - \log_4 2\)[/tex], we will use the properties of logarithms. Specifically, we can use the logarithmic property which states that [tex]\(\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)\)[/tex].
Given the expression:
[tex]\[
\log_4 32 - \log_4 2
\][/tex]
Step 1: Apply the logarithmic difference property:
[tex]\[
\log_4 32 - \log_4 2 = \log_4 \left(\frac{32}{2}\right)
\][/tex]
Step 2: Simplify the fraction inside the logarithm:
[tex]\[
\frac{32}{2} = 16
\][/tex]
So the expression becomes:
[tex]\[
\log_4 16
\][/tex]
Step 3: Evaluate [tex]\(\log_4 16\)[/tex]:
Recall that [tex]\(16\)[/tex] can be written as a power of [tex]\(4\)[/tex]:
[tex]\[
16 = 4^2
\][/tex]
Thus:
[tex]\[
\log_4 16 = \log_4 (4^2)
\][/tex]
Step 4: Use the property of logarithms [tex]\(\log_b (a^c) = c \cdot \log_b a\)[/tex]:
[tex]\[
\log_4 (4^2) = 2 \cdot \log_4 4
\][/tex]
Step 5: Recognize that [tex]\(\log_4 4 = 1\)[/tex] because any number [tex]\(a\)[/tex] raised to the power of [tex]\(1\)[/tex] is [tex]\(a\)[/tex]:
[tex]\[
2 \cdot \log_4 4 = 2 \cdot 1 = 2
\][/tex]
Therefore:
[tex]\[
\log_4 32 - \log_4 2 = 2
\][/tex]
So, the exact value of the logarithmic expression is:
[tex]\[
\boxed{2}
\][/tex]
