Which of the following is equivalent to log_(3)(b)*log_(b)(27) ? Choose 1 answer: (A) 3 (B) 9 (C) log(9) (D) log_(b)(3)

Recognize Change of Base: Recognize the use of the change of base formula. The expression log_(3)(b)*log_(b)(27) involves two logarithms with different bases. We can use the change of base formula to rewrite log_(b)(27) in terms of a common base, such as base 3. Change of Base Formula: log_(b)(A) = log_(c)(A) / log_(c)(b)Apply Formula to log: Apply the change of base formula to log_(b)(27). Using the change of base formula, we can write log_(b)(27) as log_(3)(27) / log_(3)(b). log_(b)(27) = log_(3)(27) / log_(3)(b)Simplify log_(3)(27): Simplify log_(3)(27). Since 27 is 3^3, log_(3)(27) simplifies to 3. log_(3)(27) = 3Substitute Simplified Value: Substitute the simplified value into the expression. Now we substitute log_(3)(27) = 3 into the expression from Step 2. log_(3)(b)*log_(b)(27) = log_(3)(b) * (3 / log_(3)(b))Simplify the Expression: Simplify the expression. We notice that log_(3)(b) in the numerator and denominator will cancel out, leaving us with just 3. log_(3)(b) * (3 / log_(3)(b)) = 3

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Which of the following is equivalent to
log_(3)(b)*log_(b)(27) ?
Choose 1 answer:
(A) 3
(B) 9
(C)
log(9)
(D)
log_(b)(3)

Which of the following is equivalent to $\log _{3}(b) \cdot \log _{b}(27)$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $3$ $\newline$ (B) $9$ $\newline$ (C) $\log (9)$ $\newline$ (D) $\log _{b}(3)$

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Math Problems

Algebra 2

Product property of logarithms

Full solution

Q. Which of the following is equivalent to

\log _{3}(b) \cdot \log _{b}(27)

\newline

Choose

1

answer:

\newline

(A)

3

\newline

(B)

9

\newline

(C)

\log (9)

\newline

(D)

\log _{b}(3)

Recognize Change of Base: Recognize the use of the change of base formula. The expression $\log_{3}(b)\cdot\log_{b}(27)$ involves two logarithms with different bases. We can use the change of base formula to rewrite $\log_{b}(27)$ in terms of a common base, such as base $3$ . Change of Base Formula: $\log_{b}(A) = \frac{\log_{c}(A)}{\log_{c}(b)}$
Apply Formula to log: Apply the change of base formula to $\log_{b}(27)$ . Using the change of base formula, we can write $\log_{b}(27)$ as $\frac{\log_{3}(27)}{\log_{3}(b)}$ . $\log_{b}(27) = \frac{\log_{3}(27)}{\log_{3}(b)}$
Simplify $\log_{3}(27)$ : Simplify $\log_{3}(27)$ . $\newline$ Since $27$ is $3^3$ , $\log_{3}(27)$ simplifies to $3$ . $\newline$ $\log_{3}(27) = 3$
Substitute Simplified Value: Substitute the simplified value into the expression. $\newline$ Now we substitute $\log_{3}(27) = 3$ into the expression from Step $2$ . $\newline$ $\log_{3}(b)\cdot\log_{b}(27) = \log_{3}(b) \cdot \left(\frac{3}{\log_{3}(b)}\right)$
Simplify the Expression: Simplify the expression. $\newline$ We notice that $\log_{3}(b)$ in the numerator and denominator will cancel out, leaving us with just $3$ . $\newline$ $\log_{3}(b) \times (\frac{3}{\log_{3}(b)}) = 3$