Chapter 5: Exponential and Logarithmic Functions 5.6: Solving Exponential Logarithmic Equations Day 1 Essential Question: Give examples of equations that.

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


Powers of the Different Bases◦Solve the equation 5x = 2

5x = 2 log52 = x

log 2/log 5 = x x = 0.4307


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


Using Substitution◦Set u back to ex, and solve

2 5

ln(2 5) ln(2 5)

1.4436

xe

x x undefined

x


Assignment◦Page 386◦Problems 1-31, odd problems◦Show work



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


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


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


Assignment◦Page 386◦Problems 53-67, odd problems◦Show work



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


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


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


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


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


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


Assignment◦Page 386◦Problems 35-51 & 69-75, odd

problems◦Show work

Chapter 5: Exponential and Logarithmic Functions 5.6: Solving Exponential Logarithmic Equations Day 1 Essential Question: Give examples of equations that.

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