To find [tex]\(\log _8 0.87\)[/tex] using the change-of-base theorem, follow these steps:
1. Understand the Change-of-Base Formula:
The change-of-base formula for logarithms 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]
This formula allows us to change the base of the logarithm to something more convenient, typically base 10 (common logarithms) or base [tex]\(e\)[/tex] (natural logarithms).
2. Apply the Change-of-Base Formula:
We need to find [tex]\(\log_8(0.87)\)[/tex]. Using base 10 for convenience, we apply the change-of-base formula:
[tex]\[
\log_8(0.87) = \frac{\log_{10}(0.87)}{\log_{10}(8)}
\][/tex]
3. Calculate [tex]\(\log_{10}(0.87)\)[/tex] and [tex]\(\log_{10}(8)\)[/tex]:
- [tex]\(\log_{10}(0.87) \approx -0.06069784035361199\)[/tex]
- [tex]\(\log_{10}(8) \approx 0.9030899869919435\)[/tex]
4. Divide the Logarithms:
[tex]\[
\log_8(0.87) = \frac{-0.06069784035361199}{0.9030899869919435} \approx -0.067
\][/tex]
5. Simplify and Round the Answer:
The exact value of [tex]\(\log_8(0.87)\)[/tex] is approximately [tex]\(-0.06697089797533214\)[/tex]. When rounding to four decimal places, the result is:
[tex]\[
\log_8(0.87) \approx -0.0670
\][/tex]
Thus, the simplified and rounded value of [tex]\(\log_8(0.87)\)[/tex] is:
[tex]\[
\log_8(0.87) = -0.0670
\][/tex]
