Method

Solving Logarithmic Equations Algebraically

If only one side of a logarithmic equation can be written as a single logarithm with base

b,

with

b > 0

and

b \neq = 1,

then the equation can be solved by using the definition of a logarithm.

lo g_{b} m = n \Leftrightarrow b^{n} = m

Since logarithms are defined only for positive numbers,

m

must be a positive number. Consider an example equation.

4 lo g_{2} (x - 1) + 6 = 14

To solve this equation, four steps must be followed.

Isolate the Logarithmic Expression on One Side

The first step is to isolate the logarithmic expression. This can be done by using inverse operations.

4 lo g_{2} (x - 1) + 6 = 14

$LHS - 6 = RHS - 6$

4 lo g_{2} (x - 1) = 8

$LHS / 4 = RHS / 4$

lo g_{2} (x - 1) = 2

Use the Definition of a Logarithm

Once the logarithmic expression is isolated, the definition of a logarithm can be used to eliminate the logarithm.

lo g_{2} (x - 1) = 2 \Leftrightarrow 2^{2} = x - 1

Solve the Obtained Equation

The obtained equation can now be solved.

2^{2} = x - 1

▼

Solve for

x

Calculate power

4 = x - 1

$LHS + 1 = RHS + 1$

5 = x

Rearrange equation

x = 5

Check the Solutions

Finally, to detect any extraneous solutions, the obtained value can be checked by substituting it into the original equation.

4 lo g_{2} (x - 1) + 6 = 14

$x = 5$

4 lo g_{2} (5 - 1) + 6 = ? 14

▼

Evaluate left-hand side

Subtract term

4 lo g_{2} 4 + 6 = ? 14

$lo g_{c} a = \frac{lo g _{b} a}{lo g _{b} c}$

4 (\frac{lo g 4}{lo g 2}) + 6 = ? 14

Use a calculator

4 (2) + 6 = ? 14

8 + 6 = ? 14

Add terms

14 = 14 ✓

A true statement was obtained, so the solution is not extraneous.