Sure, let's find [tex]\(\log_4(5)\)[/tex] using the change-of-base theorem.
The change-of-base theorem states that for any positive numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] where [tex]\( b \neq 1 \)[/tex] and [tex]\( c \neq 1 \)[/tex]:
[tex]\[
\log_b(a) = \frac{\log_c(a)}{\log_c(b)}
\][/tex]
In this problem, we're looking to find [tex]\(\log_4(5)\)[/tex]. We can choose any base [tex]\(c\)[/tex] for our logarithms, but it’s common to use the base 10 (common logarithm) or base [tex]\(e\)[/tex] (natural logarithm). Here, we'll use base 10.
Applying the change-of-base theorem:
[tex]\[
\log_4(5) = \frac{\log_{10}(5)}{\log_{10}(4)}
\][/tex]
Now, we need to find the values of [tex]\(\log_{10}(5)\)[/tex] and [tex]\(\log_{10}(4)\)[/tex].
By calculation:
[tex]\[
\log_{10}(5) \approx 0.6989700043360187
\][/tex]
[tex]\[
\log_{10}(4) \approx 0.6020599913279623
\][/tex]
So, substituting these values into the formula:
[tex]\[
\log_4(5) = \frac{0.6989700043360187}{0.6020599913279623} \approx 1.1609640474436813
\][/tex]
Therefore, [tex]\(\log_4(5) \approx 1.1609640474436813\)[/tex].
