Every real logarithm has a positive base different from 1 and a positive argument. For \(\log_a b=1/\log_b a\), also require \(b\ne1\). Any root index is a positive integer. Formulas labelled wrong are not identities; special values can still give equality.
Because logarithms are indices, the index laws translate into three log laws. They apply only when every logarithm has the same base and every argument is positive.
Laws of logarithms (same base \(a\), positive arguments)
| Law | Statement | Example |
|---|---|---|
| Product law | \(\log_a(MN)=\log_a M+\log_a N\) | \(\log_2(5x)=\log_2 5+\log_2 x\) |
| Quotient law | \(\log_a\dfrac{M}{N}=\log_a M-\log_a N\) | \(\ln\dfrac{x}{x+4}=\ln x-\ln(x+4)\) |
| Power law | \(\log_a M^n=n\log_a M\) | \(\lg x^5=5\lg x\) \(\lg\sqrt[3]{x}=\dfrac13\lg x\) |
Watch the brackets
| Wrong | Correct |
|---|---|
| \(\log_a(x+y)=\log_a x+\log_a y\) | The product law splits \(\log_a(xy)\) only. A logarithm of a sum cannot be split. |
| \(\lg 2x^3=3\lg 2x\) | The index 3 sits on \(x\) only: \(\lg 2x^3=\lg 2+3\lg x\) \(\lg(2x)^3=3\lg 2x\) |
| \((\log_a x)^n=n\log_a x\) | The power law needs the power inside the argument: \(\log_a(x^n)=n\log_a x\). \((\log_a x)^n\) is a power of the whole logarithm. |
Change of base
| Form | Statement | Example |
|---|---|---|
| New base \(c\) | \(\log_a b=\dfrac{\log_c b}{\log_c a}\) | \(\log_9 x=\dfrac{\log_3 x}{\log_3 9}=\dfrac{\log_3 x}{2}\) |
| Reciprocal | \(\log_a b=\dfrac{1}{\log_b a}\) | \(\log_x 3=\dfrac{1}{\log_3 x}\) |
| Calculator bases | \(\log_a b=\dfrac{\lg b}{\lg a}=\dfrac{\ln b}{\ln a}\) | \(\log_5 18=\dfrac{\lg 18}{\lg 5}\) \(=1.80\ (3\text{ s.f.})\) |
Log laws combine logarithms of the SAME base only. If the bases differ, change the base first, then apply the laws.
Need help? Join our JC Math tuition classes.
Learn more