What you need to know
Logarithms turn powers into multipliers and products into sums. Most exam questions become manageable once the log laws are applied carefully and domain restrictions are checked.
- Interpret a logarithm as an exponent.
- Use product, quotient and power laws in the correct direction.
- Check that every logarithm argument is positive.
- Convert between logarithmic and exponential forms when solving equations.
Key mathematics
Definition
\log_a b=c\iff a^c=b
Here a is positive, a is not 1, and b is positive.Product law
\log_a(xy)=\log_a x+\log_a y
Products become sums.Power law
\log_a(x^k)=k\log_a x
Bring the exponent in front.Worked example
Solve $\log_2(x-1)+\log_2(x+1)=3$. Combine logs: $$\log_2(x^2-1)=3.$$ Hence $x^2-1=8$, so $x=\pm3$. The domain requires $x-1>0$, therefore the valid solution is $x=3$.
EXAM TECHNIQUE
Exam tips
- Write domain restrictions before solving when the variable is inside a logarithm.
- Combine logs before converting to exponential form when possible.
- Reject algebraic roots that make any original log argument non-positive.
WATCH OUT
Common mistakes
- Using $\log(x+y)=\log x+\log y$; there is no such sum law.
- Keeping a solution that violates the log domain.
- Confusing the base of the logarithm with the argument.
Frequently asked questions
What is the fastest way to solve a log equation?
Usually combine logarithms using the laws, then rewrite the equation in exponential form.
Why are domain restrictions important?
Real logarithms are defined only for positive arguments.
Is there a log law for addition?
No. The standard laws apply to products, quotients and powers, not sums.