Properties of Logarithms

By the end of this lesson, you will be able to apply the product, quotient, and power properties to expand and condense logarithmic expressions, and use the change-of-base formula to evaluate logarithms of any base on a calculator.

The Big Idea

Logarithms transform multiplicative relationships into additive ones — that's their historical purpose (simplifying calculations before computers). The three core properties formalize this transformation and are the tools used to solve virtually all logarithmic equations. Mastering expansion and condensing is the gateway to solving.

The Three Properties

Product Property: $$\log_b(MN) = \log_b M + \log_b N$$

Quotient Property: $$\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$$

Power Property: $$\log_b(M^p) = p \cdot \log_b M$$

All three hold for $M, N > 0$ and any valid base $b$.

Expanding Logarithms

"Expanding" means breaking a single log into a sum/difference of simpler logs.

Example 1 — Expanding

Expand: $\log_3\left(\dfrac{x^2 \sqrt{y}}{z^3}\right)$

$$= \log_3(x^2) + \log_3(\sqrt{y}) - \log_3(z^3)$$

$$= 2\log_3 x + \frac{1}{2}\log_3 y - 3\log_3 z$$

Condensing Logarithms

"Condensing" (writing as a single log) reverses expansion. Apply the properties in reverse:

• Sum → product property
• Difference → quotient property
• Coefficient → power property

Example 2 — Condensing

Condense: $3\ln x - 2\ln y + \ln z$

$$= \ln(x^3) - \ln(y^2) + \ln z$$

$$= \ln\left(\frac{x^3}{y^2}\right) + \ln z = \ln\left(\frac{x^3 z}{y^2}\right)$$

Example 3 — Numerical Application

Given $\log 2 \approx 0.301$ and $\log 3 \approx 0.477$, find $\log 72$.

$72 = 8 \times 9 = 2^3 \times 3^2$

$\log 72 = 3\log 2 + 2\log 3 = 3(0.301) + 2(0.477) = 0.903 + 0.954 = 1.857$

Change-of-Base Formula

To compute $\log_b x$ using a calculator (which only has $\log$ or $\ln$):

$$\log_b x = \frac{\log x}{\log b} = \frac{\ln x}{\ln b}$$

Example: $\log_5 30 = \dfrac{\log 30}{\log 5} = \dfrac{1.477}{0.699} \approx 2.113$