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Apply the product law of logarithms

For a fixed logarithm base b>0b>0, b≠1b\ne1, the logarithm of a product of positive quantities satisfies log⁡b(MN)=log⁡bM+log⁡bN\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)\log_b(M+N) can be separated, and this scope is limited to real logarithms with positive arguments, not complex or more advanced generalizations.

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⁡bM+log⁡bN\log_b(MN)=\log_b M+\log_b N

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

Example: Expand log⁡2(8x)\log_2(8x), where x>0x>0.

  1. Identify the product inside the logarithm:

8x=8⋅x 8x=8\cdot x

  1. Apply the product law:

log⁡2(8x)=log⁡28+log⁡2x \log_2(8x)=\log_2 8+\log_2 x

  1. Evaluate log⁡28\log_2 8. Since 23=82^3=8,

log⁡28=3 \log_2 8=3

  1. Write the final result:

log⁡2(8x)=3+log⁡2x \boxed{\log_2(8x)=3+\log_2 x}

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

log⁡b(M+N)≠log⁡bM+log⁡bN.\log_b(M+N)\ne \log_b M+\log_b N.

Learn by doing: Apply the product law of logarithms

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    Logarithms - Product Property - Sum to Product (Variables)

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