Laws of Logarithms II

Logarithmic laws provide essential tools for simplifying and evaluating exponential expressions, particularly in mathematical and applied settings where powers and repeated multiplication play a central role. Two important rules are the power law and the change-of-base formula, both allowing for transforming expressions into more manageable forms.

The power law of logarithms states that the logarithm of a number raised to an exponent equals the exponent multiplied by the logarithm of the base number. In symbolic form, this is written as

This rule is especially useful in real-world contexts such as compound interest, where an initial amount grows over time through repeated multiplication. To solve for time, the equation is typically rearranged by dividing both sides by the principal, isolating the growth factor. Taking logarithms of both sides sets up the expression for simplification, and the power law allows the exponent, which often represents time, to be extracted from the exponent position and solved as a linear variable.

The change-of-base formula enables the evaluation of logarithms with arbitrary bases by expressing them in terms of logarithms with a base supported by common calculators, such as base 10 or base e. The formula is written as

For example, calculating log⁡₄(50) becomes

This conversion allows accurate computation using standard logarithmic functions available on most calculators.