Question

for any positive numbers a, b, and d, with b ≠ 1, \log_{b}\underline{quadquad} = d \bullet \log_{b} a.\
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a. \log_{b} a + \log_{b} d\
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b. d^{a}\
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c. a^{d} \bullet \log_{b} a^{d}\
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d. a^{d}

Explanation:

Step1: Recall Logarithm Power Rule

The power rule of logarithms states that for any positive numbers \(a\), \(b\) (where \(b
eq1\)) and any real number \(d\), \(\log_{b}(a^{d})=d\cdot\log_{b}a\).

Step2: Match with Given Equation

We are given that \(\log_{b}(\text{blank}) = d\cdot\log_{b}a\). By the power rule of logarithms, the blank should be \(a^{d}\) since \(\log_{b}(a^{d})=d\cdot\log_{b}a\) matches the given equation.

Answer:

D. \(a^d\)