For x > 0, a > 0, and a ≠ 1

Properties of Logarithms

Common Errors

There is no formula to simplify these expressions:

Solving Exponential Equations –
An exponential equation is an equation with the variable in an exponent.

1. Isolate the exponential expression (the base and exponent).

Ex: 100 ×1.023x = 200   Divide both sides by 100
1.023x = 2

2. Take the logarithm ( log or ln ) of both sides and bring down the exponent using
property 7 on the other side of this page.

Ex: ln1.023x = ln2
3x ln1.02 = ln2

3. Solve for the variable and evaluate using a calculator if necessary .

Solving Logarithmic Equations

1. Move all terms containing logarithms to one side of the equation, and all other
terms to the other side of the equation.

2. Combine the terms with logarithms to get a single logarithm with a coefficient
of 1 using properties 5, 6, and 7 (work from the right side to the left of each

Property 7
Property 6

3. Rewrite the equation in exponential form.

4. Solve for the variable and check all solutions in the original equation.

When checking, -6 does not give a solution since the domain of all logarithmic
functions is x > 0. Therefore, the solution is x = 6.

NOTE: These procedures work for most equations, but additional techniques such as
factoring may be required. Not all exponential and logarithmic equations are possible
to solve.

Prev Next