Which of the following is equivalent to $\log (a) \cdot \log _{a}(5)$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\log (a)$ $\newline$ (B) $\log (5)$ $\newline$ (C) $\log (5 a)$ $\newline$ (D) $\log _{a}(5 a)$

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Algebra 2

Product property of logarithms

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Q. Which of the following is equivalent to

\log (a) \cdot \log _{a}(5)

\newline

Choose

1

answer:

\newline

(A)

\log (a)

\newline

(B)

\log (5)

\newline

(C)

\log (5 a)

\newline

(D)

\log _{a}(5 a)

Understanding the expression: Understand the expression $\log(a)\cdot\log_{a}(5)$ . $\newline$ The expression consists of two logarithms being multiplied together: $\log(a)$ , which is the common logarithm of $a$ , and $\log_{a}(5)$ , which is the logarithm base $a$ of $5$ .
Recognizing applicable logarithm properties: Recognize the properties of logarithms that might apply. $\newline$ The multiplication of two logarithms does not directly correspond to any of the basic logarithm properties (product, quotient, or power rules). However, we can interpret $\log_{a}(5)$ as the power to which we must raise $a$ to get $5$ .
Applying the definition of logarithm base $a$ of $5$ : Apply the definition of the logarithm base $a$ of $5$ . $\newline$ By definition, $\log_{a}(5) = x$ means that $a^x = 5$ . Therefore, $\log_{a}(5)$ is the exponent $x$ .
Substituting the definition back into the original expression: Substitute the definition back into the original expression. $\newline$ Since $\log_{a}(5)$ is the exponent $x$ that makes $a^x = 5$ , we can rewrite the expression as $\log(a) \cdot x$ .
Recognizing the expression cannot be further simplified: Recognize that the expression $\log(a) \cdot x$ does not simplify further using common logarithm properties. $\newline$ The expression $\log(a) \cdot x$ is already in its simplest form, given that $x$ is the exponent that satisfies $a^x = 5$ . There is no property of logarithms that allows us to combine a logarithm and an exponent in this way.
Matching the expression to the given options: Match the expression to the given options. $\newline$ None of the options $(A) \log(a)$ , $(B) \log(5)$ , $(C) \log(5a)$ , or $(D) \log_{a}(5a)$ are equivalent to the expression $\log(a) \cdot x$ . Therefore, none of the options are equivalent to the original expression $\log(a)\cdot\log_{a}(5)$ .