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]
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]
