Skill: Apply the product law of logarithms

Explanation and Free Practice Resources

For a fixed logarithm base $b>0$ , $b\ne1$ , the logarithm of a product of positive quantities satisfies $\log_b(MN)=\log_b M+\log_b N$ , allowing products to be expanded into sums or sums of logarithms to be combined into a product. The relationship reflects multiplication of powers with a common base; it does not mean that $\log_b(M+N)$ can be separated, and this scope is limited to real logarithms with positive arguments, not complex or more advanced generalizations.

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Detailed Explanation: Apply the product law of logarithms

For the same logarithm base, a product inside a logarithm can be rewritten as a sum:

\log_b(MN)=\log_b M+\log_b N

This works when $M>0$ and $N>0$ . The base must satisfy $b>0$ and $b\ne 1$ .

Example: Expand $\log_2(8x)$ , where $x>0$ .

Identify the product inside the logarithm:

8x=8\cdot x

Apply the product law:

\log_2(8x)=\log_2 8+\log_2 x

Evaluate $\log_2 8$ . Since $2^3=8$ ,

\log_2 8=3

Write the final result:

\boxed{\log_2(8x)=3+\log_2 x}

Remember, this law applies to multiplication, not addition. In general,

\log_b(M+N)\ne \log_b M+\log_b N.

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Logarithm Algebra (Product Property) - To Answer (Coefficient 1)

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