Skill: Apply the quotient law of logarithms

Explanation and Free Practice Resources

For positive real numbers $M$ and $N$ , and a logarithm base $b>0,\ b\ne1$ , the quotient law represents division inside a logarithm as subtraction: $\log_b(M/N)=\log_b M-\log_b N$ . This supports rewriting and simplifying logarithmic expressions and solving related equations; the quotient is not $\log_b M/\log_b N$ , and complex-valued arguments or more advanced generalizations are outside this scope.

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Detailed Explanation: Apply the quotient law of logarithms

When a logarithm contains a quotient, rewrite the division as subtraction:

\log_b\left(\frac{M}{N}\right)=\log_b M-\log_b N

This works when $M>0$ , $N>0$ , and the base satisfies $b>0$ and $b\ne 1$ . Be careful: it is not

\frac{\log_b M}{\log_b N}.

Example

Simplify:

\log_2\left(\frac{32}{4}\right)

Step 1: Apply the quotient law.

\log_2\left(\frac{32}{4}\right)=\log_2 32-\log_2 4

Step 2: Evaluate each logarithm.

Since $2^5=32$ and $2^2=4$ ,

\log_2 32=5 \qquad\text{and}\qquad \log_2 4=2

Step 3: Subtract.

5-2=3

Therefore,

\boxed{\log_2\left(\frac{32}{4}\right)=3}

The quotient law changes division inside a logarithm into subtraction between two logarithms.

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Logarithms - Quotient Property - Division as Fraction To Difference (Variables)

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