Question
Let \( b=\log k \). Write the following expression in terms of \( b \). Assume \( k>0 \). \( \log \left(k^{5}\right)=\square \)
Ask by Harper Lane. in the United States
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression \( \log \left(k^{5}\right) \) in terms of \( b \) is \( 5b \).
Solution
1. We are given \( b = \log k \).
2. Using the power rule of logarithms, we have:
\[
\log \left(k^5\right) = 5 \log k
\]
3. Substitute \( \log k \) with \( b \):
\[
5 \log k = 5b
\]
Thus, the expression in terms of \( b \) is \( 5b \).
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Simplify this solution Extra Insights
To rewrite the expression in terms of \( b \), we can use the power rule of logarithms, which states that \( \log(a^n) = n \log(a) \). In this case, we apply the rule to the given expression: \[ \log \left(k^{5}\right) = 5 \log(k) \] Since \( b = \log(k) \), we can substitute \( b \) into the equation: \[ \log \left(k^{5}\right) = 5b \] Thus, the expression \( \log \left(k^{5}\right) \) can be written as \( 5b \).
