* Use like bases to solve exponential equations. * Use logarithms to solve exponential equations. * Use the definition of a logarithm to solve logarithmic equations. * Use the one-to-one property of logarithms to solve logarithmic equations. * Solve applied problems involving exponential and logarithmic equations.

1. 6.6 Exponential & Logarithmic
Equations
Chapter 6 Exponential and Logarithmic Functions

2. Concepts and Objectives
⚫ Objectives for this section are
⚫ Use like bases to solve exponential equations.
⚫ Use logarithms to solve exponential equations.
⚫ Use the definition of a logarithm to solve logarithmic
equations.
⚫ Use the one-to-one property of logarithms to solve
logarithmic equations.
⚫ Solve applied problems involving exponential and
logarithmic equations.

3. Exponential Equations
⚫ Exponential equations are equations with variables as
exponents.
⚫ If you can re-write each side of the equation using a
common base, then you can set the exponents equal to
each other and solve for the variable.
⚫ Example: Solve =
1
5
125
x
−
= 3
5 5
x
= −3
x
= 3
125 5

4. Exponential Equations (cont.)
⚫ Example: Solve + −
=
1 3
3 9
x x

5. Exponential Equations (cont.)
⚫ Example: Solve + −
=
1 3
3 9
x x
= 2
9 3
( )
−
+
=
3
1 2
3 3
x
x
( )
+ = −
1 2 3
x x
+ = −
1 2 6
x x
=
7 x
( )
−
+
=
2 3
1
3 3
x
x

6. Exponential Equations (cont.)
⚫ To solve an equation with exponents, you can “undo” the
exponent by raising each side to the reciprocal.
⚫ Solve =
5 2
243
b

7. Exponential Equations (cont.)
⚫ To solve an equation with exponents, you can “undo” the
exponent by raising each side to the reciprocal.
⚫ Solve =
5 2
243
b
( ) ( )
=
2
2
5 5
5
2
243
b
= 9
b

8. Using Logs to Solve Equations
⚫ If you can’t rewrite the equation to a common base, you
can use logarithms and the power property of logs to
solve the equation.
⚫ Take the log of both sides. You can use either the
common logarithm (base 10) or natural logarithm. (I
usually use the natural log because it’s one less letter to
write.)
⚫ Use the power property to move the exponent in front of
the log.

9. Exponential Equations
⚫ Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
− +
=
2 1 2
3 0.4
x x

10. Exponential Equations
⚫ Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
− +
=
2 1 2
3 0.4
x x
− +
=
2 1 2
log3 log0.4
x x

11. Exponential Equations
⚫ Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
− +
=
2 1 2
3 0.4
x x
− +
=
2 1 2
log3 log0.4
x x
( ) ( )
=
− +
2 1 2
log3 log0.4
x x

12. − = +
2 log3 log3 log0.4 2log0.4
x x
Exponential Equations
⚫ Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
− +
=
2 1 2
3 0.4
x x
− +
=
2 1 2
log3 log0.4
x x
( ) ( )
=
− +
2 1 2
log3 log0.4
x x

13. Exponential Equations
⚫ Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
− +
=
2 1 2
3 0.4
x x
− +
=
2 1 2
log3 log0.4
x x
( ) ( )
− = +
2 1 log3 2 log0.4
x x
− = +
log0.
l 4
og3
2 log3 2log0.4
x x
− = +
log0.4
2 log3 2log g
0.4 lo 3
x x

14. Exponential Equations
⚫ Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
− +
=
2 1 2
3 0.4
x x
− +
=
2 1 2
log3 log0.4
x x
( ) ( )
− = +
2 1 log3 2 log0.4
x x
− = +
2 log3 log3 log0.4 2log0.4
x x
( )
− = +
2log3 log0.4 2log0.4 log3
x
− = +
2 log3 log0.4 2log0.4 log3
x x

15. Exponential Equations
⚫ Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
− +
=
2 1 2
3 0.4
x x
− +
=
2 1 2
log3 log0.4
x x
( ) ( )
− = +
2 1 log3 2 log0.4
x x
− = +
2 log3 log3 log0.4 2log0.4
x x
( )
− = +
2log3 log0.4 2log0.4 log3
x
− = +
2 log3 log0.4 2log0.4 log3
x x
+
=  −
−
2log0.4 log3
0.236
2log3 log0.4
x

16. Properties of Logs, Revisited
⚫ You could also finish this as
+
=
−
2log0.4 log3
2log3 log0.4
x
+
=
−
2
2
log0.4 log3
log3 log0.4
( )
=
 
 
 
log 0.16 3
9
log
0.4
=
log0.48
log22.5
This is an
exact answer.

17. Solving a Logarithmic Equation
⚫ Example: Solve ( ) ( )
log 6 log 2 log
x x x
+ − + =

18. Solving a Logarithmic Equation
⚫ Example: Solve ( ) ( )
log 6 log 2 log
x x x
+ − + =
6
log log
2
x
x
x
+
=
+
6
2
x
x
x
+
=
+
( )
6 2
x x x
+ = +
2
6 2
x x x
+ = +
2
6 0
x x
+ − =
( )( )
3 2 0
x x
+ − =
3, 2
x = −
= + −
2
0 6
x x
If we plug in –3, x+2
is negative. We can’t
take the log of a
negative number, so
our answer is 2.

19. Solving a Base e Equation
⚫ Example: Solve and round your answer to the
nearest thousandth.
2
200
x
e =

20. Solving a Base e Equation
⚫ Example: Solve and round your answer to the
nearest thousandth.
2
200
x
e =
2
ln ln200
x
e =
2
ln200
x =
ln200
x = 
2.302
x  

21. Solving a Base e Equation
⚫ Example: Solve ( )
ln
ln ln 3 ln2
x
e x
− − =

22. Solving a Base e Equation
⚫ Example: Solve ( )
ln
ln ln 3 ln2
x
e x
− − =
( )
− − =
ln ln 3 ln2
x
x
( )
− − =
ln
ln ln 3 ln2
x
x
e
ln ln2
3
x
x
=
−
2
3
x
x
=
−
2 6
x x
= −
6 x
=
( )
= −
2 3
x x

23. Classwork
⚫ College Algebra 2e
⚫ 6.6: 4-10, 12-28 (4); 6.5: 16-24 (even); 6.3: 44-56
(4), 64, 66
⚫ 6.6 Classwork Check
⚫ Quiz 6.5