Evaluate $\\log_{25}125$ Using Logarithm Power Rules

$Math Problem$

Mathematics Grade 10 (Senior High School)

Question Content

Evaluate $\\log_{25}125$

Correct Answer

$\\frac{3}{2}$ or 1.5

Detailed Solution Steps

1
Step 1: Rewrite the base and the argument using the same prime base. We know $25=5^2$ and $125=5^3$.
2
Step 2: Substitute into the logarithm: $\\log_{25}125=\\log_{5^2}5^3$.
3
Step 3: Apply the change of base power rule for logarithms: $\\log_{a^m}b^n=\\frac{n}{m}\\log_{a}b$. When $a=b$, $\\log_{a}a=1$, so this simplifies to $\\frac{n}{m}$.
4
Step 4: Here, $a=5$, $m=2$, $n=3$. So $\\log_{5^2}5^3=\\frac{3}{2}\\log_{5}5=\\frac{3}{2}\\times1=\\frac{3}{2}$.

Knowledge Points Involved

Prime Factorization for Logarithms

This technique involves rewriting the base and argument of a logarithm as powers of a common prime number. It is used to simplify logarithms with non-prime bases or arguments, making it easier to apply logarithmic rules.

Logarithm Power Rule for Bases and Arguments

The rule $\\log_{a^m}b^n=\\frac{n}{m}\\log_{a}b$ (for $a>0,a\\neq1,b>0,m\\neq0$) allows adjusting the exponents of the base and argument of a logarithm. When the base and argument are powers of the same number, this rule simplifies the logarithm to a rational number by canceling out the common base's logarithm (since $\\log_{a}a=1$).

Basic Logarithm Value $\\log_{a}a=1$

This states that the logarithm of a number with the same base is always 1, because $a^1=a$. It is a fundamental logarithmic property used to simplify final calculations after applying other rules.

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