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← Solving Logarithmic Equations and Inequalities

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.

1

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

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

3

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

4

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

Multiply

8 + 6 = ? 14

Add terms

14 = 14 ✓

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

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