1. Solve a logarithmic equation
EXAMPLE 4
Solve a logarithmic equation
Solve log (4x – 7) = log (x + 5). 5
SOLUTION
log (4x – 7) = log (x + 5). 5
Write original equation.
4x – 7 = x + 5
Property of equality for logarithmic equations
3x – 7 = 5
Subtract x from each side.
3x = 12
Add 7 to each side.
x = 4
Divide each side by 3.
The solution is 4.
ANSWER

2. Solve a logarithmic equation
EXAMPLE 4
Solve a logarithmic equation
Check: Check the solution by substituting it into the original equation.
(4x – 7) = (x – 5)
log 5
Write original equation.
(4 4 – 7) = (4 + 5) ?
log 5
Substitute 4 for x.
9 =
log 5
Solution checks.

3. Exponentiate each side of an equation
EXAMPLE 5
Exponentiate each side of an equation
Solve (5x – 1)= 3
log 4
SOLUTION
(5x – 1)=
(5x – 1)= 3
log 4
Write original equation.
4log4(5x – 1) = 4 3
Exponentiate each side using base 4.
b = x
log b x
5x – 1 = 64
5x = 65
Add 1 to each side.
x = 13
Divide each side by 5.
The solution is 13.
ANSWER

4. Exponentiate each side of an equation
EXAMPLE 5
Exponentiate each side of an equation
log 4
(5x – 1) = (5 13 – 1) =
Check:
log 4
Because 4 = 64, = 3. 3

5. Standardized Test Practice
EXAMPLE 6
Standardized Test Practice
SOLUTION
log 2x + log (x – 5) = 2
Write original equation.
log [2x(x – 5)] = 2
Product property of logarithms
= 10
log 2
[2x(x – 5)]
Exponentiate each side using base 10.
2x(x – 5) = 100
Distributive property

6. Standardized Test Practice
EXAMPLE 6
Standardized Test Practice
2x – 10x = 100 2
b = x
log b x
2x – 10x – 100 = 0 2
Write in standard form.
x – 5x – 50 = 0 2
Divide each side by 2.
(x – 10)(x + 5) = 0
Factor.
x = 10 or x = – 5
Zero product property
Check: Check the apparent solutions 10 and –5 using algebra or a graph.
Algebra: Substitute 10 and –5 for x in the original equation.

7. EXAMPLE 6
Standardized Test Practice
log 2x + log (x – 5) = 2
log 2x + log (x – 5) = 2
log (2 10) + log (10 – 5) = 2
log [2(–5)] + log (–5 – 5) = 2
log 20 + log 5 = 2
log (–10) + log (–10) = 2
log 100 = 2
Because log (–10) is not defined, –5 is not a solution.
2 = 2
So, 10 is a solution.

8. EXAMPLE 6
Standardized Test Practice
Graph: Graph y = log 2x + log (x – 5) and y = 2 in the same coordinate plane. The graphs intersect only once, when x = 10. So, 10 is the only solution.
The correct answer is C.
ANSWER

9. Solve the equation. Check for extraneous solutions.
GUIDED PRACTICE
for Examples 4, 5 and 6
Solve the equation. Check for extraneous solutions.
7. ln (7x – 4) = ln (2x + 11)
9. log 5x + log (x – 1) = 2
SOLUTION 5
SOLUTION 3
8. log (x – 6) = 5 2
log (x + 12) + log x =3 4
SOLUTION 4
SOLUTION 38