Skip to main content

Exponent Laws and Logarithmic Properties

There are several rules that are helpful when working with exponential functions.

Law of Exponents:

Law of Exponents

The first law states that to multiply two exponential functions with the same base, we simply add the exponents. The second law states that to divide two exponential functions with the same base, we subtract the exponents. The third law states that in order to raise a power to a new power, we multiply the exponents. The fourth and fifth laws state that in order to raise a product or quotient to a power, we raise each factor to that power.

Example: Simplify the expression:

law of exponents example

Example: Simplify the expression:

law of exponents example

Note: We cannot simplify any further since the terms in the numerator and denominator do not have the same base.

Log Laws

There are three properties that are useful when working with logarithmic functions.

Properties of Logarithms

If x, y > 0 and r is any real number, then

loga(xy) = loga x + logay

loga(x/y) = loga x - logay

loga(xr) = rlogax

Example: Simplify the expression log25 + log23

Solution:

log25 + log23

= log2(5 x 3)

= log2(15)

Example: Simplify the expression 5(In2)

5(In2)

= In(25)

= In(32)

Exponent Laws Example:

University of Ontario Institute of Technology logo