Download Exponential and Logarithmic Equations

7-5 Logarithmic & Exponential
Equations
Terms and Concepts
these will be on a quiz
EXAMPLE 1
Solve by equating exponents
x
x–3
1
2
Solve 4 =
SOLUTION
1
2
x
4 =
x–3
x
x–3
(22 ) = (2 ) – 1
2x
2 = 2
–x+3
2x = –x + 3
x=1
ANSWER
The solution is 1.
Write original equation.
1
Rewrite 4 and 2 as powers with
base 2.
Power of a power property
Property of equality for
exponential equations
Solve for x.
for Example 1
GUIDED PRACTICE
Solve the equation.
1. 9
2x
= 27
x–1
–3
SOLUTION
2. 100
7x + 1
SOLUTION
3. 81
= 1000
3x – 2
–8
5
3–x
=
SOLUTION
1 5x – 6
3
–6
EXAMPLE 2
Take a logarithm of each side
Solve 4x = 11.
SOLUTION
4 x= 11
Write original equation.
log 4x = log 4 11
Take log of each side.
4
4
x = log 11
4
ANSWER
log b b x= x
log 11
x=
log 4
Change-of-base formula
x
Use a calculator.
1.73
The solution is about 1.73. Check this in
equation.
the original
GUIDED PRACTICE
for Examples 2 and 3
Solve the equation.
x
4. 2 = 5
SOLUTION
about 2.32
9x
5. 7 = 15
SOLUTION
about 0.155
Terms and Concepts
these will be on a quiz
EXAMPLE 4
Solve a logarithmic equation
Solve log 5(4x – 7) = log 5(x + 5).
SOLUTION
log 5(4x – 7) = log 5 (x + 5).
Write original equation.
4x – 7 = x + 5
Property of equality for
logarithmic equations
3x – 7 = 5
Subtract x from each side.
3x = 12
x=4
ANSWER
Add 7 to each side.
Divide each side by 3.
The solution is 4.
EXAMPLE 5
Exponentiate each side of an equation
Solve log 4(5x – 1)= 3
SOLUTION
log (5x – 1)= 3
4
4log4(5x – 1) = 4 3
5x – 1 = 64
5x = 65
x = 13
ANSWER
Write original equation.
Exponentiate each side using
base 4.
blogbx = x
Add 1 to each side.
Divide each side by 5.
The solution is 13.
GUIDED PRACTICE
for Examples 4, 5 and 6
Solve the equation. Check for extraneous solutions.
7. ln (7x – 4) = ln (2x + 11)
SOLUTION
8.
3
log 2(x – 6) = 5
SOLUTION
38
9. log 5x + log (x – 1) = 2
SOLUTION
10.
5
log 4(x + 12) + log x4=3
SOLUTION
4