5-4 Exponential and Logarithmic Equations

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
2. 5x-2 = 25x
3. 6(3x+1) = 54
4. e–x2
= e-3x - 4

5. Exponentials of Unequal Bases
• Use logarithm (inverse function) of same
base on both sides of equation
EX: Solve: ex = 72  lnex = ln72 xlne = ln72
x = ln72 (calc ready form) x ~ 4.277
EX: Solve: 7x-1 = 12 log77x-1 = log712
(x-1)log77 = log712  x-1 = log712
x = 1+log712 x ~ 1.277

6. You try…
1. Solve 3(2x) = 42
2. Solve 32t-5 = 15
3. Solve e2x = 5
4. Solve ex + 5 = 60

7. Solving Logarithmic Equations
• Convert to exponential (inverse) form
EX: Solve: lnx = -1/2
elnx = e-1/2  x = e-1/2  x  .607
EX: Solve: 2log53x = 4  log53x = 2 (get the log by itself)
52 = 3x  25/3 = x
Use Properties of Logs to condense
EX: Solve: log4x + log4(x-1) = ½  log4(x2-x)= ½
41/2 = x2 – x  0 = x2-x-2 (x-2)(x+1) x=2

8. You try…
1. Solve lnx = -7
2. Solve 2log5 3x = 4
3. Solve . lnx+ln(x-3) = 1
4. Solve . 5 + 2ln x = 4

9. Double-Sided Log Equations
• Equate powers (domain solutions only)
EX: Solve: log5(5x-1) = log5(x+7)
5x – 1 = x + 7  x = 2
EX: Solve: ln(x-2) + ln(2x-3) = 2lnx
Use a property: ln(x-2)(2x-3) = lnx2
2x2 – 7x + 6 = x2  x2-7x+6=0  x = 6 & 1

10. You try…
1. Solve ln3x2 = lnx
2. Solve log6(3x + 14) – log6 5 = log6 2x
3. Solve log2x+log2(x+5) = log2(x+4)

11. 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!