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Logarithms

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.
Ex:

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.
Ex:

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).

Ex:
Property 7
Property 6

3. Rewrite the equation in exponential form.
Ex:

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

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.

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