OBJECTIVES
1. Solve Logarithmic Equations Using the Properties of Logarithms
2. Solve Exponential Equations
Objective 1: Solve Logarithmic Equations Using Properties of Logarithms
Steps:
1. Use properties of logs to make a singular logarithm (if necessary)
2. Distribute
3. Change to exponential form (y=logak to x=ak)
4. Use the quadratic formula or factoring (Zero Product Property) to solve
5. Make sure the argument is positive when verifying the solution (logak=x, k is the argument)
Example 1:
ln(x)+ln(x+2)=4
ln(x(x+2))=4 *property “loga+logb=logab”
ln(x2+2x)=4 *distributive property
e4=x2+2x *change to exponential form (e is the base of ln)
0=x2+2x-e4 *set equal to zero
Solution: {6.4564}
Use the quadratic formula to solve for “x”
(-2+√(4+4e4))/2 = 6.4564
(-2-√(4+4e4))/2 = -8.4564
Only 6.4564 is a solution because this solution makes the argument positive
Example 2:
log4(x+3)+log4(2-x)=1
log4((x+3)(2-x))=1 *property “loga+logb=logab”
log4(2x+6-x2-3x)=1 *distribute
log4(-x2-x+6)=1 *simplify
41=-x2-x+6 *change to exponential form
0=-x2-x+2 *set equal to zero
x2+x-2=0
(x-1)(x+2)=0 *factor
x=1,-2 *Solve by Zero Product Property
Solutions: {-2, 1}
Both solutions make the argument positive
Example 3:
log7(3x+8)=log7(x2+2x-4)
3x+8=x2+2x-4 *both sides can be divided by log7
0=x2-x-12 *simplify
0=(x-4)(x+3) *factor
x=4,-3 *solve using Zero Product Property
Solution: {4}
Only 4 is a solution to this equation because -3 makes the argument negative
Example 4:
log(5x+1)-log(2x-3)=2
log10((5x+1)/(2x-3))=2 *property “loga-logb=loga/b”
102=(5x+1)/(2x-3)*change to exponential form
100/1=(5x+1)/(2x-3) *make a proportion
200x-300=5x+1 *cross multiply
195x=301 *solve for “x”
x=301/195
Solution: {301/195}
There is only one solution and it makes the argument positive
Objective 2: Solve Exponential Equations
The natural logarithm and properties of logarithms can be used to solve exponential equations.
Example 1:
7x=15
ln7x=ln15
xln7=ln15
x(ln7)/(ln7)=(ln15)(ln7)
x=(ln15)/(ln7)
X≈.95582290
Example 2:
8x-5=67x+5
ln8x-5=ln67x+5 *take the ln of both sides
(x-5)ln8=(7x+5)ln6 *use “logab=bloga” property
(ln8)x-5ln8=(7ln6)x+5ln6 *distribute
(ln8)x-(7ln6)x=5ln8+5ln6 *put all terms with “x” on the left
(ln8-7ln6)x=5(ln8+ln6) *factor out the common factor
x=(5(ln8+ln6))/(ln8-7ln6)
x≈-1.84997006
Example 3:
5+3*62x=18
3*62x=13 *subtract 5 from both sides
62x=13/3 *divide both sides by 3
ln62x=ln(13/3) *take ln of both sides
2xln6=ln(13/3) *use “logab=bloga” property
xln6=(ln(13/3))/2 *divide both sides by 2
x(ln6)/(ln6)=((ln(13/3))/2)/(ln6)
x=((ln(13/3))/2)/(ln6)
x≈.40918915
Practice Problems:
Solve the following logarithmic equations analytically:
log5(4x+11)=2
log2(x+5)-log2(2x-1)=5
log8x+log8(x+6)=log8(5x+12)
log4(3x-2)-log4(4x+1)=2
log3(x2-6x)=3
Solve the following exponential equations analytically:
75x+10=73x-4
81x-4=93x
4-x+2 =19
e2x+3=40
8(2)x-5=36