Quick Revision Notes — Logarithm Laws & Properties
60-Second Summary
The core takeaways in under a minute
Logarithmic properties expand and compress equations: the Product Rule converts products to sums ($\log(MN) = \log M + \log N$), the Quotient Rule converts fractions to differences ($\log(M/N) = \log M - \log N$), and the Power Rule drops exponents in front ($k\log M$). Common base-10 logarithms ($\log x$) are used in earthquake Richter scales, decibel loudness, and pH scales. Natural logarithms ($\ln x$) use base $e \approx 2.718$. Remember: logarithm base $b$ must satisfy $b > 0, b \neq 1$, and input $a > 0$.
Logarithms properties matrix
Rules to memorize for calculations
| Law / Property | Formula | Key application |
|---|---|---|
| Product Law | $\log_a(MN) = \log_a(M) + \log_a(N)$ | Collapsing separate logs under addition |
| Quotient Law | $\log_a(M/N) = \log_a(M) - \log_a(N)$ | Simplifying log subtraction terms |
| Power Law | $\log_a(M^k) = k\log_a(M)$ | Evaluating powers of numbers |
| Change of Base | $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$ | Converting variable bases to standard base 10 |
| Log of Base | $\log_a(a) = 1$ | Evaluating base elements |
| Log of 1 | $\log_a(1) = 0$ | Simplifying 1 values |
Common Mistakes to Avoid
Watch out for these classic exam traps!
$\log_a(M + N) \neq \log_a(M) + \log_a(N)$. There is NO property to distribute logs over addition. The correct rule is for products: $\log_a(MN) = \log_a(M) + \log_a(N)$.
$\frac{\log_a(M)}{\log_a(N)} \neq \log_a(M - N)$. The quotient rule is: $\log_a\left(\frac{M}{N}\right) = \log_a(M) - \log_a(N)$. The fraction of two logs is evaluated using the change of base formula: $\frac{\log_a(M)}{\log_a(N)} = \log_N(M)$.
Writing $\log_a(M) = \frac{\log_b(a)}{\log_b(M)}$. Remember that the base of the original log ($a$) goes to the denominator: $\log_a(M) = \frac{\log_b(M)}{\log_b(a)}$.