Exponentials and logarithms

What you’ll learn

• Use logarithm laws with valid domains.
• Solve exponential and logarithmic equations.

Understand the idea

A logarithm answers an exponent question: $\log_a b=c$ means $a^c=b$. Its base must be positive and different from 1, and its argument must be positive. For $a>1$, $a^x$ grows as $x$ increases; for $0<a<1$, it decays. Both pass through $(0,1)$.

Logarithm laws convert products into sums, quotients into differences and powers into multipliers. They do not split a sum inside a logarithm. When solving, combine logarithms first if useful, change to exponential form, and check every candidate against the original positive-argument conditions.

Key formulae

$$\log_a(xy)=\log_a x+\log_a y$$

$$\log_a(x/y)=\log_a x-\log_a y$$

$$\log_a(x^k)=k\log_a x\quad(x>0)$$

$$a^x=b\Rightarrow x=\frac{\log b}{\log a}$$

Worked example

Solve $\log_2(x-1)+\log_2(x+1)=3$.

1. The domain is $x>1$. Combine the logs: $\log_2((x-1)(x+1))=3$.
2. Exponentiate: $x^2-1=2^3=8$, so $x^2=9$.
3. The candidates are $x=3$ and $x=-3$; only $3$ satisfies the original domain.

Answer: $x=3$.