Upload
vernon-boone
View
224
Download
2
Tags:
Embed Size (px)
Citation preview
5-4 Exponential & 5-4 Exponential & Logarithmic EquationsLogarithmic Equations
Strategies and PracticeStrategies and Practice
ObjectivesObjectives– 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.
Use like bases to solve Use like bases to solve exponential equationsexponential 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
You try… 1. 4x = 83
2. 5x-2 = 25x
3. 6(3x+1) = 54
4. e–x2 = e-3x - 4
(22)x = (23)3 (2)2x = (2)9
So 2x = 9, or x = 4.5
5x-2 = (52)x 5x-2 = (5)2x So x - 2 = 2x, or x = -
2 3x+1 = 9 3x+1 = 32
So x + 1= 2, or x = 1
So –x2 = -3x - 4 & x2 – 3x – 4 = 0 & (x-4)(x+1) = 0 & x=4 and x = -1
Exponentials of Unequal BasesExponentials of Unequal Bases• Use logarithm (inverse function) of same base
on both sides of equation
Solve: ex = 72
ex = 72
loge logex = ln 72 4.277
Solve: 7x-1 = 12 7x-1 = 12log7 log7
x - 1 = log7 12x = log7 12 + 1x = + 1
log 12log 7
2.277
You try…You try…1. Solve 3(2x) = 42
2. Solve 32t-5 = 15
3. Solve e2x = 5
4. Solve ex + 5 = 60
x = log2 14 3.807
t = 1/2(log3 15 + 5) 3.732
x = 1/2 ln 5 0.805
x = ln 55 4.007
Solving Logarithmic EquationsSolving Logarithmic Equations• Rewrite into exponential form
EX: Solve: ln x = - 1/2
loge x = - 1/2
e -1/2 = xx = e -1/2
EX: Solve: 2 log5 3x = 4 log5 3x = 2
52= 3x 25= 3x
25/3= x x = 25/3
0.607
8.333
Solving Logarithmic EquationsSolving Logarithmic Equations• Use properties of logarithms to condense.
EX: Solve: log4x + log4(x-1) = ½log4 x(x – 1) = 1/2
4 1/2 = x(x – 1)2 = x2 – x0 = x2 – x – 20 = (x – 2)(x + 1)x = 2 & x = -1
Check for extraneous roots.
You try…You try…1. Solve ln x = -7
2. Solve 2 log3 2x = 4
3. Solve ln x + ln (x-3) = 0
4. Solve 5 + 2ln x = 4
x = e-7 0.000912
x = 9/2
x = & x = 3 + 132
3 - 132
x = e-1/2 0.607
Double-Sided Log Equations• Equate powers (domain solutions only)
EX: Solve: log5(5x-1) = log5(x+7) 5x – 1 = x +
7 4x = 8x = 2
EX: Solve: ln(x-2) + ln(2x-3) = 2lnx
Use the properties to condense.
ln (x-2)(2x-3) = ln x2
(x-2)(2x-3) = x2
2x2 – 7x + 6 = x2
x2 – 7x + 6 = 0 (x – 6)(x – 1)= 0
x = 6 & x = 1
Check for extraneous roots.
x = -2 + 22 & x = -2 - 2 2
You try…
1. Solve ln3x2 = lnx
2. Solve log6(3x + 14) – log6 5 = log6 2x
3. Solve log2x+log2(x+5) = log2(x+4)
x = 0 & x = 1/3
x = 2