# Logarithms

For x > 0, a > 0, and a ≠ 1

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 ×1.02^{3x} = 200 Divide both sides by 100

1.02^{3x} = 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: ln1.02^{3x} = 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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