1. 7-5 Logarithmic & Exponential Equations

2. Terms and Concepts these will be on a quiz

3. Write original equation.
EXAMPLE 1
Solve by equating exponents
Solve 4 = x 1 2
x – 3
SOLUTION
4 = x 1 2
x – 3
Write original equation.
Rewrite 4 and as powers with base 2. 1 2
(2 ) = (2 ) 2
x – 3 x
– 1
2 = 2 2x
– x + 3
Power of a power property
2x = –x + 3
Property of equality for exponential equations
x = 1
Solve for x.
The solution is 1.
ANSWER

4. GUIDED PRACTICE for Example 1 Solve the equation. 3. 81 = 1 3=
3 – x 1 3
5x – 6
= 27 2x
x – 1
SOLUTION –3
SOLUTION –6
= 1000
7x + 1
3x – 2
– 8 5
SOLUTION

5. Write original equation.
EXAMPLE 2
Take a logarithm of each side
Solve 4 = 11. x
SOLUTION
4 = 11 x
Write original equation.
log 4x = log 4 11 4
Take log of each side. 4
log 4
x =
log b = x b x
x =
log 11
log 4
Change-of-base formula x
Use a calculator.
The solution is about Check this in the original equation.
ANSWER

6. GUIDED PRACTICE for Examples 2 and 3 Solve the equation. 4. 2 = 5
= 5 x
SOLUTION
about 2.32
= 15 9x
SOLUTION
about

7. Terms and Concepts these will be on a quiz

8. 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

9. 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

10. 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