Solving Exponential and Logarithmic Equations

Solving when the exponent has a variable.

Rules of exponents

• Multiply……add

• Divide………subtract

• Power to a power….multiply

x0 =1

x−1 =1x

x =x12

Radicals become Fractional exponents:

power

rootx23 =x

23

Practice with exponents

• See textbook pg. 292

Radical expressions

• Express the when coverting a radical

• Expression with a fractional exponent.

• Example:

• 1. What is the power?

• 2. what is the root?

power

root

835

Fractional exponents

• Express as

•

power

root

835= 8

53

Now evaluate:

= 32

Practice:

• See page 296 in textbook.do 3-6

• 297 20, 21 46-48, 58, 68,69

• Do these:1. 44

2

2. 73

2

Do now:

• Express with fractional exponents:

• Solve:

x + 3

x + 3 =2

Solving when the base is a variable:

• Solve the previous problem:

(x + 3)12 =2

((x+ 3)12 )2 =22

x+ 3=4x=1

Procedure:

• Isolate the exponential, raise to the reciprocal power, and solve:

4x3

2 =108

Procedure:

• Isolate the exponential, raise to the reciprocal power, and solve:

4x3

2 =108

x32 =27

(x32 )

23 =27

23

x=9

Example:

• Isolate first!

3x3 −2 =22

Example:

• Isolate first!

3x3 −2 =22

3x3 =24

x3 =8

x=813 =2

• One way to solve exponential equations is to use the property that if 2 powers w/ the same base are equal, then their exponents are equal.

• if bx = by, then x=y• For b>0 & b≠1

Exponential Equations

base 2, base 3,base 5 numbers:

• base 2: base 3: base 5:

• 2 3 5

• 4 9 25

• 8 27 125

• 16 81

• 32

• 64

Solve by equating exponents

• 43x = 8x+1

• But 4 = 22 and 8 = 23

• (22)3x = (23)x+1 rewrite w/ same base

Solve by equating exponents

• 43x = 8x+1

• (22)3x = (23)x+1 rewrite w/ same base

• 26x = 23x+3 power to a power: multiply

• 6x = 3x+3 set exponents equal

• x = 1 Check → 43*1 = 81+1

64 = 64

Procedure:

• Copy these steps

• 1. decide like base

• 2. rewrite one or both numbers as a power of the base

• 3. simplify the exponents

• 4. set exponents equal and solve

• 5. be sure to check!

Your turn!

• 24x = 32x-1

Solution

• 24x = 32x-1

• 24x = (25)x-1

• 4x = 5x-5

• 5 = x

Be sure to check your answer!!!

Hand this one in:

• Solve and check:

34 x =27x+1

Hand this one in:

• Solve and check:

Graphs of exponentials

• Growth and decay:

• growth decay

Interest rate formula:

• A=a0(1+r)t

Exponential growth and decay formula:

y = abx

a is the initial value b is used to find the growth or decay rate.x is the time If b >1 then b - 1 is the growth rateIf b < 1 then 1 – b is the decay rate

Note: the rate is always in decimal form.

Interest rate

• Use the formula to determine the amount of money you will have after 3 years if you invest 100 at a rate of 8%.

• A=a0(1+r)t

Interest rate

• Use the formula to determine the amount of money you will have after 3 years if you invest 100 at a rate of 8%.

a =a0 (1+ r)t

a=100(1+.08)t

a=100(1.08)3

a=125.97

Exponential examples

• Y = 300(.95)x

• What is the initial value?

• What is the rate of decay?

• How much will there be in 10 years?

• Y = 20(1.3)t

• What is the initial value

• What is the growth rate?

• How much will it be in 5 years?

COMPOUND INTEREST FORMULA

amount at the end

Principal(amount at start)

annual interest rate

(as a decimal)

A =a0 1+rn

⎛

⎝⎜

⎞

⎠⎟

nttime(in

years)

number of times per year that interest in

compounded

Calculate how much you will have ifYou invest 100 at 8% compounded Quarterly after 3 years.

A =a0 1+rn

⎛

⎝⎜

⎞

⎠⎟

nt

Calculate how much you will have ifYou invest 100 at 8% compounded Quarterly after 3 years.

A =a0 1+rn

⎛

⎝⎜

⎞

⎠⎟

nt

a=100(1+.084

)4(3)

a=100(1.02)12

a=126.82

Compound or CONTINUOUS growth:The rate is changed to a decimal if not already.

Ex: population grows continuously at a rate of 2% in Allentown. If Allentown has 10,000 people today, how many will it have 5 yearsFrom now?

A =a0ert

Compound or CONTINUOUS growth:

Ex: population grows continuously at a rate of 2% in Allentown. If Allentown has 10,000 people today, how many will it have 5 yearsFrom now?

A =a0ert

A =a0ert

A=10,000e5(.02)

A=10000e.1 ≈11052

How much will a 1000 investment be Worth in 10 years if it is compounded continuouslyat 2.5%?

If a substance decays continually at a rate of -0.05 an hour,How much will be left of a 20 gram substance in 6 hours?

How much will a 1000 investment be Worth in 10 years if it is compounded continuouslyAt 2.5%?

If a substance decays continually at a rate of -0.05 an hour,How much will be left of a 20 gram substance in 6 hours?

A =a0ert

A=1000e10(.025)

A=1284.03

A =a0ert

A=20e6(−.05)

A=14.82

Solving exponentials w/ variable exponents-like bases

• 1. decide like base

• 2. rewrite one or both numbers as a power of the base

• 3. simplify the exponents

• 4. set exponents equal and solve

• 5. be sure to check!

• Example 43x = 8x+1

Solve by equating exponents

• 43x = 8x+1

• (22)3x = (23)x+1 rewrite w/ same base

• 26x = 23x+3 power to a power: multiply

• 6x = 3x+3 set exponents equal

• x = 1 Check → 43*1 = 81+1

64 = 64

When you can’t rewrite using the same base:

1. isolate the base term2. “log” both sides

3. Solve for x

• Example: 2x = 7

• log 2x = log 7

When you can’t rewrite using the same base, you can solve by taking a log

of both sides

• 2x = 7

• log 2x = log 7

• x log 2 = log 7

• x = ≈ 2.8072log

7log

4x = 15“log” both sides & solve

4x = 15• log 4x = log 15

• x log 4 = log15

• x= log 15/log 4

• ≈ 1.95

10(2x-3) +4 = 21• Isolate the base term

10(2x-3) +4 = 21• -4 -4• 102x-3 = 17• Now solve for x….

10(2x-3) +4 = 21• -4 -4• 102x-3 = 17• log102x-3 = log17• (2x-3)(log 10) = log 17• 2x – 3 = log17/log 10• 2x =(3 +1.2304) • x= ≈ 2.115

4.2304

2

5(3)x/4 = 25• isolate and solve:

Solution:

• Ex: 5(3)x

4 =25

5(3)x

4 =25

3x4 =5

log3x4 =log5

x4

=log5log3

Solution:

• Ex: 5(3)x

4 =25

3x4 =5

log3x4 =log5

x4

=log5log3

x≈1.465(4) ≈5.860

Newton’s Law of Cooling

• The temperature T of a cooling substance @ time t (in minutes) is:

•T = (T0 – TR) e-rt + TR

• T0= initial temperature

• TR= room temperature

• r = constant cooling rate of the substance

• You’re cooking stew. When you take it off the stove the temp. is 212°F. The room temp. is 70°F and the cooling rate of the stew is r =.046. How long will it take to cool the stew to a serving temp. of 100°?

• T0 = 212, TR = 70, T = 100 r = .046

• So solve:• 100 = (212 – 70)e-.046t +70• 30 = 142e-.046t (subtract 70)

• .211 ≈ e-.046t (divide by 142)

• How do you get the variable out of the exponent?

• ln .211 ≈ ln e-.046t (take the ln of both sides)

• ln .211 ≈ -.046t ln e

• -1.556 ≈ -.046t (1)

• 33.8 ≈ t

• about 34 minutes to cool!

Cooling cont.

Domains of log equations

• Find the domain of # 19 on page 331

• Domains are important because solving logarithmic equations sometimes produces extraneous roots.

Solving Log Equations

• To solve use the property for logs w/ the same base:

• + #’s b,x,y & b≠1

• If logbx = logby, then x = y

log3(5x-1) = log3(x+7)

Solve by decompressing

log3(5x-1) = log3(x+7)

•5x – 1 = x + 7• 5x = x + 8• 4x = 8• x = 2 and check• log3(5*2-1) = log3(2+7)• log39 = log39

When you can’t rewrite both sides as logs w/ the same base exponentiate

each side

• b>0 & b≠1

•if x = y, then bx = by

log5(3x + 1) = 2

• use base 5 on both sides

log5(3x + 1) = 2

• (3x+1) = 52

• 3x+1 = 25

• x = 8 and check

• Because the domain of log functions doesn’t include all reals, you should check for extraneous solutions

log5x + log(x+1)=2• Decompress and exponentiate

log5x + log(x+1)=2• log (5x)(x+1) = 2 (product property)

• log (5x2 + 5x) = 2

• 10log5x +5x = 102

• 5x2 + 5x = 100

• x2 + x - 20 = 0 (subtract 100 and divide by 5)

• (x+5)(x-4) = 0 x=-5, x=4• graph and you’ll see 4=x is the only solution

2

One More!

log2x + log2(x-7) = 3Solve and check:

One More!

log2x + log2(x-7) = 3• log2x(x-7) = 3• log2 (x2- 7x) = 3• 2log

2(x -7x) = 23

• x2 – 7x = 8• x2 – 7x – 8 = 0• (x-8)(x+1)=0• x=8 x= -1

2

Page 331 # 38

• Solve and ck.

• log(x-2) + log(x+5) =2log 3

Solution:

• log(x-2) + log(x+5) =2log 3

• (x-2)(x+5)=9

• x2 + 3x -19 = 0

• (the neg. answer is extraneous)

x =−3+ 85

2≈3.1

Assignment

Hw pg 331 18-25, 33-37 odds