The value of $\log_5 \frac{(125)(625)}{25}$ is equal to

Concept & Formula Simplify the argument of the logarithm first by expressing all terms with a common base, then apply the logarithm power rule: $$ \log_a (a^n) = n $$ Step-by-Step Solution * **Given:** Evaluate $\log_5 \frac{(125)(625)}{25}$. * **Calculation:** First, simplify the fraction. Express all numbers as powers of $5$: $125 = 5^3$ $625 = 5^4$ $25 = 5^2$ * Substitute these into the fraction: $$ \frac{5^3 \cdot 5^4}{5^2} $$ * Apply the laws of exponents ($x^a \cdot x^b = x^{a+b}$ and $\frac{x^a}{x^b} = x^{a-b}$): $$ \frac{5^{3+4}}{5^2} = \frac{5^7}{5^2} = 5^{7-2} = 5^5 $$ * Now, substitute this simplified value back into the original logarithm: $$ \log_5 (5^5) $$ * The base and the logarithm cancel out, leaving the exponent: $$ 5 $$ Exam Strategy & Shortcut Using logarithmic addition/subtraction rules directly is slower. Simplifying the exponential terms first as powers of $5$ ($3 + 4 - 2 = 5$) gives the final answer in a single mental step. Common Pitfall Multiplying $125 \times 625$ manually creates unnecessarily large numbers ($78125$), wasting precious exam time and increasing the chance of arithmetic errors. Always use exponent rules. Final Answer **Therefore, the correct answer is 5.**