Select the correct answer.

Which exponential equation is equivalent to the logarithmic equation [tex]\(\log_x 5 + \log_x 12 = 7\)[/tex]?

A. [tex]\(x^7 = 17\)[/tex]

B. [tex]\(x^7 = 60\)[/tex]

C. [tex]\(7^x = 60\)[/tex]

D. [tex]\(7^x = 17\)[/tex]



Answer :

To solve the given logarithmic equation [tex]\(\log_x 5 + \log_x 12 = 7\)[/tex], let's follow these steps:

1. Use the properties of logarithms:
According to the properties of logarithms, specifically the product rule for logarithms, we can combine the two logarithms on the left-hand side:
[tex]\[ \log_x 5 + \log_x 12 = \log_x (5 \cdot 12) \][/tex]
So the equation becomes:
[tex]\[ \log_x (5 \cdot 12) = 7 \][/tex]

2. Simplify inside the logarithm:
Calculate the product inside the logarithm:
[tex]\[ 5 \cdot 12 = 60 \][/tex]
Thus, the equation simplifies to:
[tex]\[ \log_x 60 = 7 \][/tex]

3. Rewrite the logarithmic equation in exponential form:
To convert the logarithmic equation [tex]\(\log_x 60 = 7\)[/tex] to its equivalent exponential form, recall that [tex]\(\log_x 60 = 7\)[/tex] means:
[tex]\[ x^7 = 60 \][/tex]

Thus, the exponential equation that is equivalent to the logarithmic equation [tex]\(\log_x 5 + \log_x 12 = 7\)[/tex] is:
[tex]\[ x^7 = 60 \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{B. \, x^7 = 60} \][/tex]