chapter 5: exponential and logarithmic functions 5.6: solving exponential logarithmic equations day...
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Chapter 5: Exponential and Logarithmic Chapter 5: Exponential and Logarithmic FunctionsFunctions5.6: Solving Exponential Logarithmic 5.6: Solving Exponential Logarithmic EquationsEquationsDay 1Day 1Essential Question: Give examples of equations that can be solved by using the properties of exponents and logarithms.
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Powers of the Same Base◦Solve the equation 8x = 2x+1
8x = 2x+1
(23)x = 2x+1
23x = 2x+1
Set the exponents equal to each other
3x = x+1 2x = 1 x = 1/2
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Powers of the Different Bases◦Solve the equation 5x = 2
5x = 2 log52 = x
log 2/log 5 = x x = 0.4307
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Powers of the Different Bases◦Solve the equation 24x-1 = 31-x
◦Take one base and make it into a log problem log231-x = 4x-1
(1 – x)log23 = 4x-1
(1 – x)(log 3/log 2) = 4x – 1
(1 – x)(1.5850) = 4x – 1Calculate log 3/log 2
1.5850 – 1.5850x = 4x – 1 Distribute on left
2.5850 – 1.5850x = 4x Add 1 to both sides
2.5850 = 5.5850x Add 1.5850x to both sides
x = 0.4628 Divide by 5.5850
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquationsUsing Substitution
◦Solve the equation ex – e-x = 4 ex – e-x = 4
Multiply all terms by ex to remove the negative exponent
e2x – 1 = 4ex
Set everything equal to 0, substitute u = ex
e2x – 4ex – 1 = 0 u2 – 4u – 1 = 0 This is now a…
Quadratic Equation
22 ( 4) ( 4) 4(1)( 1)4
2 2(1)
4 16 4 4 20 4 2 52 5
2 2 2
b b acu
a
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Using Substitution◦Set u back to ex, and solve
2 5
ln(2 5) ln(2 5)
1.4436
xe
x x undefined
x
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Assignment◦Page 386◦Problems 1-31, odd problems◦Show work
Chapter 5: Exponential and Logarithmic Chapter 5: Exponential and Logarithmic FunctionsFunctions5.6: Solving Exponential Logarithmic 5.6: Solving Exponential Logarithmic EquationsEquationsDay 2Day 2Essential Question: Give examples of equations that can be solved by using the properties of exponents and logarithms.
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Applications of Exponential Equations◦Radiocarbon Dating◦The half-life of carbon-14 is 5730
years, so the amount of carbon-14 remaining at time t is given by Many of these problems will deal with
percentage of carbon-14 remaining, so P = 1 (i.e. 100%), and the amount remaining will be the percentage left.
5730( ) (0.5)t
M t P
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Applications: Carbon Dating◦The skeleton of a mastodon has lost
58% of its original carbon-14. When did the mastodon die? If 58% has been lost, then 42% remains 5730
0.5 5730
log0.42log0.5 5730
log0.42log0.5
0.42 (0.5)
log 0.42
(5730)
7171.3171
t
t
t
t
t
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Applications: Compound Interest◦If $3000 is to be invested at 8% per
year, compounded quarterly, in how many years will the investment be wroth $10,680?
◦ 40.084
4106803000
1.02
log3.56log1.02
(1 )
10680 3000(1 )
1.02
log 3.56 4
4
64.12 4 16.03
ntrn
t
t
A P
t
t
t t
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Assignment◦Page 386◦Problems 53-67, odd problems◦Show work
Chapter 5: Exponential and Logarithmic Chapter 5: Exponential and Logarithmic FunctionsFunctions5.6: Solving Exponential Logarithmic 5.6: Solving Exponential Logarithmic EquationsEquationsDay 3Day 3Essential Question: Give examples of equations that can be solved by using the properties of exponents and logarithms.
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Applications: Population Growth◦A culture started at 1000 bacteria. 7
hours later, there are 5000 bacteria. Find the function and when there are 1 billion bacteria. Function is based off A = Pert. Need to
find r. 7
7
5000 1000
5
ln 5 7
0.2299
rt
r
r
A Pe
e
e
r
r
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Applications: Population Growth◦To find A=1,000,000, need to find t◦
0.2299
0.2299
1,000,000,000 1000
1,000,000
ln1,000,000 0.2299
60.0936 hours
rt
t
t
A Pe
e
e
t
t
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Logarithmic Equations◦Solve the equation
ln(x – 3) + ln(2x + 1) = 2(ln x)◦ln[(x – 3)(2x + 1)] = ln x2
◦ln(2x2 – 5x – 3) = ln x2
Natural logs cancel each other out
◦2x2 – 5x – 3 = x2
◦x2 – 5x – 3 = 0 Use quadratic equation
◦ 5 37
2x
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Logarithmic Equations◦Solve the equation
ln(x – 3) + ln(2x + 1) = 2(ln x)◦
◦Because = -0.5414, it’s undefined for ln(x – 3), so there’s only one solution
5 37
2x
5 37
2
5 37
2
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Equations with logarithmic & constant terms◦Solve ln(x – 3) = 5 – ln(x – 3)◦ln(x – 3) + ln(x – 3) = 5◦2 ln(x – 3) = 5◦ln (x – 3) = 2.5◦e2.5 = x – 3◦e2.5 + 3 = x◦x = 15.1825
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Equations with logarithmic & constant terms◦Solve log(x – 16) = 2 – log(x – 1)◦ log(x – 16) + log(x – 1) = 2◦ log [(x – 16)(x – 1)] = 2◦ log (x2 – 17x + 16) = 2◦102 = x2 – 17x + 16◦0 = x2 – 17x – 84◦0 = (x – 21)(x + 4)◦x = 21 or x = -4
x = -4 would give log(-4 – 16) = log -20, which is undefined
There is only one solution, x = 21
5.6: Solving Exponential and Logarithmic 5.6: Solving Exponential and Logarithmic EquationsEquations
Assignment◦Page 386◦Problems 35-51 & 69-75, odd
problems◦Show work