Properties of Logarithms
Properties of Logarithms
The properties of logarithms transform complicated expressions into simpler ones. The product rule, quotient rule, and power rule let you break a single logarithm into a sum or difference, or combine multiple logarithms into one — mirroring the laws of exponents in reverse.
These properties are not just algebraic curiosities. They are essential tools for solving exponential and logarithmic equations, and they underpin the mathematics of sound (decibels), earthquakes (Richter scale), and information theory (entropy).
This lesson drills all three properties and the change-of-base formula, which lets you evaluate logarithms of any base using a standard calculator.
These three properties are the foundation for simplifying and solving logarithmic equations.
| Property | Rule |
|---|---|
| Product Rule | $\log_b(MN) = \log_b(M) + \log_b(N)$ |
| Quotient Rule | $\log_b\!\left(\dfrac{M}{N}\right) = \log_b(M) - \log_b(N)$ |
| Power Rule | $\log_b(M^p) = p \cdot \log_b(M)$ |
Expanding Logarithmic Expressions
Expand $\log_2(8x)$.
$$\log_2(8x) = \log_2(8) + \log_2(x) = 3 + \log_2(x)$$
Expand $\log\!\left(\dfrac{x^3}{y^2}\right)$.
$$\log\left(\frac{x^3}{y^2}\right) = \log(x^3) - \log(y^2) = 3\log(x) - 2\log(y)$$
Expand $\ln\!\left(x^2 \sqrt{y}\right)$.
$$\ln(x^2 y^{1/2}) = 2\ln(x) + \tfrac{1}{2}\ln(y)$$
Condensing Logarithmic Expressions
Condense $2\log(x) + 3\log(y)$.
$$\log(x^2) + \log(y^3) = \log(x^2 y^3)$$
Condense $\log_3(5) + \log_3(4) - \log_3(10)$.
$$\log_3\!\left(\frac{5 \cdot 4}{10}\right) = \log_3(2)$$
Change of Base Formula
$$\log_b(x) = \frac{\log(x)}{\log(b)} = \frac{\ln(x)}{\ln(b)}$$
This lets you evaluate any logarithm using a calculator (which only has $\log$ and $\ln$ keys).
Evaluate $\log_5(30)$ using the change-of-base formula.
$$\log_5(30) = \frac{\ln(30)}{\ln(5)} = \frac{3.4012}{1.6094} \approx 2.113$$
Practice Problems
Show Answer Key
1. $\log(5) + \log(x) + \log(y)$
2. $4\ln(a) - 3\ln(b)$
3. $3\log_2(x) + \log_2(y) + \frac{1}{2}\log_2(z)$
4. $\log(21)$
5. $\ln\!\left(\frac{x^3}{y}\right)$
6. $\log\!\left(\frac{\sqrt{x} \cdot y^3}{z}\right)$
7. $\frac{\log(50)}{\log(7)} \approx 2.012$
8. False! $\log(a+b) \ne \log(a) + \log(b)$. The product rule says $\log(ab) = \log(a) + \log(b)$.
9. $2 - 2\log(x)$
10. $\log_4\!\left(\frac{x^2 z}{y^{1/3}}\right)$
11. $\log(12) = \log(4 \cdot 3) = 2\log(2) + \log(3) = 0.602 + 0.477 = 1.079$
12. $3 + 2 = 5$