1. 5-4 Exponential & Logarithmic Equations
Strategies and Practice

2. Objectives
– 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.

3. Use like bases to solve exponential equations
Equal bases must have equal exponents
EX: Given 3x-1 = 32x + 1 then x-1 = 2x+1 x = -2
If possible, rewrite to make bases equal
EX: Given 2-x = 4x+1 rewrite 4 as 22
2-x = 22x+2 then –x=2x+2  x=-2/3
Note:
Isolate function if needed 3(2x)=48 2x =16

4. You try… 1. 4x = 83
(22)x = (23)3
(2)2x = (2)9
So 2x = 9, or x = 4.5
2. 5x-2 = 25x
5x-2 = (52)x
5x-2 = (5)2x
So x - 2 = 2x, or x = -2
3. 6(3x+1) = 54
3x+1 = 9
3x+1 = 32
So x + 1= 2, or x = 1
4. e–x2 = e-3x - 4
So –x2 = -3x & x2 – 3x – 4 = 0 & (x-4)(x+1) = 0 & x=4 and x = -1

5. Exponentials of Unequal Bases
Use logarithm (inverse function) of same base on both sides of equation
Solve: ex = 72
loge
ex =
loge
x = ln 72
 4.277
Solve: x-1 = 12
log7
7x-1 =
log7
x - 1 = log7 12
x = log
log 12
log 7
 2.277
x =

6. You try… 1. Solve 3(2x) = 42 x = log2 14  3.807 2. Solve 32t-5 = 15
t = 1/2(log )  3.732
3. Solve e2x = 5
x = 1/2 ln 5  0.805
4. Solve ex + 5 = 60
x = ln 55  4.007

7. Solving Logarithmic Equations
Rewrite into exponential form
EX: Solve: ln x = - 1/2
loge x = - 1/2
e -1/2 = x
 0.607
x = e -1/2
EX: Solve: log5 3x = 4
log5 3x = 2
52= 3x
25= 3x
25/3= x
x = 25/3
 8.333

8. Solving Logarithmic Equations
Use properties of logarithms to condense.
EX: Solve: log4x + log4(x-1) = ½
log4 x(x – 1) = 1/2
Check for extraneous roots.
4 1/2 = x(x – 1)
2 = x2 – x
0 = x2 – x – 2
0 = (x – 2)(x + 1)
x = 2 & x = -1

9. You try… 1. Solve ln x = -7 x = e-7  0.000912 2. Solve 2 log3 2x = 4
3. Solve ln x + ln (x-3) = 0
3 + 13 2
3 - 13 2
x = & x =
4. Solve ln x = 4
x = e-1/2  0.607

10. Double-Sided Log Equations
Equate powers (domain solutions only)
EX: Solve: log5(5x-1) = log5(x+7)
5x – 1 = x + 7
4x = 8
x = 2
EX: Solve: ln(x-2) + ln(2x-3) = 2lnx
ln (x-2)(2x-3) = ln x2
Use the properties to condense.
Check for extraneous roots.
(x-2)(2x-3) = x2
2x2 – 7x + 6 = x2
x2 – 7x + 6 = 0
(x – 6)(x – 1)= 0
x = 6 & x = 1

11. You try… 1. Solve ln3x2 = lnx x = 0 & x = 1/3
2. Solve log6(3x + 14) – log6 5 = log6 2x
x = 2
3. Solve log2x+log2(x+5) = log2(x+4)
x = 2 & x = 2

12. SUMMARY Equal bases Equal exponents
Unequal bases  Apply log of given base
Single side logs  Convert to exp form
Double-sided logs  Equate powers
Note: Any solutions that result in a log(neg) cannot be used!