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Log 27² - Log 5² / Log 5²

October 17, 2026Mathematics

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Answer

3 \log_5 3

3 steps

We will solve the expression $\frac{\log_5 27^2}{\log_5 5^2}$ .

Simplify the numerator using the logarithm property $\log_b M^k = k \log_b M$ .
$\log_5 27^2 = 2 \log_5 27$
Since $27 = 3^3$ , substitute this into the expression:
$2 \log_5 3^3$
Apply the logarithm property again:
$2 \times 3 \log_5 3 = 6 \log_5 3$
Simplify the denominator using the logarithm property $\log_b M^k = k \log_b M$ and $\log_b b = 1$ .
$\log_5 5^2 = 2 \log_5 5$
Since $\log_5 5 = 1$ :
$2 \times 1 = 2$
Divide the simplified numerator by the simplified denominator.
$\frac{6 \log_5 3}{2}$ $3 \log_5 3$