Full solution

Q. Which of the following is equivalent to

\log _{3}(a) \cdot \log (3)

\newline

Choose

1

answer:

\newline

(A)

\log (a)

\newline

(B)

\log (3)

\newline

(C)

\log (3 a)

\newline

(D)

\log _{3}(3 a)

Recognize properties of logarithms: Recognize the properties of logarithms that might be relevant. $\newline$ We are given the expression $\log_{3}(a)\cdot\log(3)$ , which involves a multiplication of two logarithms. The change of base formula might be useful here, which states that $\log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)}$ for any positive $a$ , $b$ , and $c$ ( $b, c \neq 1$ ).
Apply change of base formula to $\log(3)$ : Apply the change of base formula to $\log(3)$ . $\newline$ We can write $\log(3)$ as $\log_3(3)$ using the change of base formula, where the base $c$ is $3$ . This gives us $\log(3) = \frac{\log_3(3)}{\log_3(3)}$ .
Simplify expression from Step $2$ : Simplify the expression obtained in Step $2$ . $\newline$ Since $\log_3(3)$ is simply $1$ (because any log base $b$ of $b$ is $1$ ), the expression simplifies to $\log(3) = 1$ .
Substitute simplified expression back: Substitute the simplified expression of $\log(3)$ back into the original expression. $\newline$ We now have $\log_{3}(a)\cdot\log(3) = \log_{3}(a)\cdot 1$ .
Simplify expression from Step $4$ : Simplify the expression obtained in Step $4$ . $\newline$ Multiplying anything by $1$ leaves it unchanged, so $\log_{3}(a) \cdot 1 = \log_{3}(a)$ .
Match simplified expression to options: Match the simplified expression to the given options. $\newline$ The expression $\log_{3}(a)$ matches option (A) $\log(a)$ if we assume the base of the logarithm in option (A) is $3$ . However, this is not explicitly stated, and the base of the common logarithm is typically $10$ . Therefore, the correct match is option (D) $\log_{3}(3a)$ if we consider the base of the logarithm in option (A) to be $10$ .