Laws of Logarithms II

Logarithmic laws simplify logarithmic and exponential expressions, especially through the power law.

The power law states that the logarithm of a number raised to an exponent equals the exponent multiplied by the logarithm of that number.

This rule is especially useful when the unknown is in the exponent, such as when solving for time in compound interest, where the amount grows by repeatedly multiplying the principal by a growth factor over time based on the interest rate

Dividing both sides by the principal isolates the growth factor raised to the time exponent. Taking logarithms on both sides and applying the power law isolates the exponent, allowing time to be solved.

Another important property, the change of base formula, is derived using the quotient and power laws. It rewrites a logarithm in one base as a ratio of logarithms in another base.

This transformation is useful when a calculator only supports base 10 logarithms (log) and base e logarithms (ln).

For example, the change of base formula expresses the logarithm base 4 of 50 as the log base 10 of 50 divided by the log base 10 of 4, allowing computation with standard calculators.