3.3 Properties of Logarithms

Solving logarithmic equations requires the use of four properties.

Addition of Logarithms (the base a must be a positive number other than 1)

When adding two logarithms of equal bases, you can combine the two logs into one log with x and y being multiplied together.

Ex.

Subtraction of Logarithms (the base a must be a positive number other than 1)

When subtracting two logarithms of equal bases, you can combine the two logs into one log with x being divided by y.

Ex.

Exponents/Coefficients of Logarithms (the base a must be a positive number other than 1) 

If there is a coefficient b (which is a real number) of a log, it can be moved to be the exponent of x and vice versa.

Ex.

Change-of-Base Formula (a, b and x are positive real numbers. a and b are not 1)

If you have a log base a of x, you can change the base of the logarithm to any number that suits your needs with the log of x over the log of a. This formula is very useful for situations in which you need to calculate the numeric value of a log, but the base isn't 10 or e, which are the only two calculable on a calculator. 

Ex.