The Product Rule of Logarithms

Core Concepts

Focuses on the Product Rule, which states that the logarithm of a product is the sum of the logarithms of its factors. Includes examples of both expanding and condensing expressions.

The Product Rule of Logarithms

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If $\log_5(3) \approx 0.68$ and $\log_5(4) \approx 0.86$, what is $\log_5(12)$?

Since $12 = 3 \times 4$, $\log_5(12) = \log_5(3) + \log_5(4) \approx 0.68 + 0.86 = 1.54$.

State the Product Rule for logarithms.

For any valid base $b$, $\log_b(MN) = \log_b(M) + \log_b(N)$. The logarithm of a product is the sum of the logarithms of its factors.

Use the Product Rule to expand the expression $\log_4(7y)$.

$$\log_4(7) + \log_4(y)$$

Use the Product Rule to expand the expression $\ln(10x)$.

$$\ln(10) + \ln(x)$$

Use the Product Rule to expand $\log(100ab)$.

This can be expanded as $\log(100) + \log(a) + \log(b)$, which simplifies to $2 + \log(a) + \log(b)$.

Condense the expression $\log_2(5) + \log_2(x)$ into a single logarithm.

$$\log_2(5x)$$

Condense the expression $\ln(a) + \ln(b) + \ln(c)$ into a single logarithm.

$$\ln(abc)$$

True or False: $\log_b(M+N) = \log_b(M) + \log_b(N)$.

False. This is a common mistake. The Product Rule applies to the logarithm of a product, $\log_b(MN)$, not the logarithm of a sum.

The Product Rule of Logarithms

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