Solve the Logarithm Ratio: Finding log₈(9a)/log₈(3a)

You cannot cancel logarithms like regular fractions! $\frac{\log_8 9a}{\log_8 3a}$ is not equal to $\frac{9a}{3a}$. Use the special quotient rule for dividing logarithms instead.

This uses the quotient rule for divided logs, not the quotient property. $\log(A/B) = \log A - \log B$ is for logs of quotients, but $\frac{\log A}{\log B} = \log_B A$ is for quotients of logs!

The quotient rule only works when both logarithms have the same base. If bases differ, you'd need to use change of base formula first to make them match.

Remember: denominator becomes base, numerator becomes argument. In $\frac{\log_8 9a}{\log_8 3a}$, the bottom (3a) becomes the new base, top (9a) stays as argument.

Yes! Use the change of base formula: $\log_{3a} 9a = \frac{\log_8 9a}{\log_8 3a}$. This confirms our answer matches the original expression!