Solve for $x$: ${\log }_{25}x=3$.

• A. $x=125$
• B. $x=75$
• C. $x=15625$
• D. $x=625$

Answer: (C)

Verified step by step guidance

1. Identify the given logarithmic equation: \(\log_{25} x = 3\). This means the logarithm of \(x\) with base 25 equals 3.
2. Recall the definition of logarithm: \(\log_a b = c\) is equivalent to the exponential form \(a^c = b\). Apply this to rewrite the equation as \$25^3 = x$.
3. Express the base 25 in terms of its prime factors to simplify the calculation if needed. Since \$25 = 5^2\(, rewrite \)25^3\( as \)(5^2)^3$.
4. Use the power of a power rule: \((a^m)^n = a^{m \times n}\) to simplify \((5^2)^3\) to \$5^{2 \times 3} = 5^6$.
5. Calculate \$5^6\( to find the value of \)x$. This will give the solution to the original logarithmic equation.