3.43.4Exponential & Logarithmic EquationsExponential & Logarithmic Equations

JMerrill, 2010JMerrill, 2010

Properties of LogsProperties of Logs

Quick Review of 3.3Quick Review of 3.3

Rules of LogarithmsRules of LogarithmsIf M and N are positive real numbers and b is If M and N are positive real numbers and b is ≠ ≠ 1:1:

The Product RuleThe Product Rule:: loglogbbMN = logMN = logbbM + logM + logbbNN

(The logarithm of a product is the sum of the logarithms)(The logarithm of a product is the sum of the logarithms)

Example: log (10x) = log10 + log xExample: log (10x) = log10 + log x

You do: logYou do: log77(1000x) = (1000x) = loglog771000 + log1000 + log77xx

Rules of LogarithmsRules of LogarithmsIf M and N are positive real numbers and b If M and N are positive real numbers and b ≠ 1:≠ 1:

The Quotient RuleThe Quotient Rule

(The logarithm of a quotient is the difference of the logs)(The logarithm of a quotient is the difference of the logs)

Example: Example:

You do: You do:

log log logb b bM M NN

log log log 22x x

714log

x 7 7log 14 log x

Rules of LogarithmsRules of LogarithmsIf M and N are positive real numbers, b If M and N are positive real numbers, b ≠ 1, and p is ≠ 1, and p is

any real number:any real number: The Power Rule:The Power Rule:

loglogbbMMpp = = p p loglogbbM M (The log of a number with an exponent is the product of the (The log of a number with an exponent is the product of the

exponent and the log of that number)exponent and the log of that number)

Example: log xExample: log x22 = 2 log x = 2 log x Example: logExample: log557744 = 4 log = 4 log5577 You do: logYou do: log335599

Challenge: Challenge: log x

= = 9 log9 log335512 1log log

2 x x

CondensingCondensing Sometimes, we need to condense Sometimes, we need to condense

before we can solve: before we can solve:

Product RuleProduct Rule

Power RulePower Rule

Quotient RuleQuotient Rule

3

3

log log 3loglog 3log

log log

log

b b b

b b

b b

b

M N PMN P

MN PMNP

CondensingCondensing

Condense:Condense:

12

12

1 log log log2

log log log

log log

b b b

b b b

b b

M N P

M N P

MN MNorP P

Using the Rules to CondenseUsing the Rules to Condense

Ex:Ex:

You Do: You Do:

2

1ln20 2ln ln21ln20 ln ln2

1ln20 4 ln5

x

x

x x

2(ln2 ln ) (ln ln4) x x

24l l4

1n nxx x

2

2(ln2 ln ) ln ln4

2ln ln ln4

x x

xx

BasesBases We don’t really use other bases anymore, We don’t really use other bases anymore,

but since logs are often written in other but since logs are often written in other bases, we must change to base 10 in order bases, we must change to base 10 in order to use our calculators.to use our calculators.

Change of Base FormulaChange of Base Formula

Example Example loglog5588 ==

This is also how you graph in another base. This is also how you graph in another base. Enter Enter yy11=log(8)/log(5).=log(8)/log(5). Remember, you don’t Remember, you don’t have to enter the base when you’re in base 10!have to enter the base when you’re in base 10!

blog(c)log clog(b)

log(8) 1.290log(5)

Parentheses are vital! The log key opens the ( ), you

must close it!

3.43.4Solving Exponents & LogsSolving Exponents & Logs

Solving GuidelinesSolving Guidelines

OriginalOriginal RewrittenRewritten SolutionSolution

22xx = 32 = 32 22xx = 2 = 255 x = 5x = 5 lnx – ln3 = 0lnx – ln3 = 0 lnx = ln3lnx = ln3 x = 3x = 3 (1/3)(1/3)xx = 9 = 9 33-x-x = 3 = 322 x = -2x = -2 eexx = 7 = 7 lnelnexx = ln7 = ln7 x = ln7x = ln7 logx =-1logx =-1 1010logxlogx = 10 = 10-1-1 x = 10x = 10-1-1

= 1/10 = 1/10

Get both parts to the same base

Solve like normalGet both parts to the same

base

If you have a variable in the

exponent position, take the log of both sides. Take the ln if you’re using e,

take the log if using common logs.

If you have a log on one side,

exponentiate both sides

SolvingSolving Getting all the numbers to the same Getting all the numbers to the same

base.base. Example: Example:

3

3

128122

2 23

x

x

x

x

1

11 2

112 3 2

32 2 2

9 27

9 27

3 3

3 332 22

14

x

x

x

x

x

x

SolvingSolving

Clear the exponent:Clear the exponent:32

32

223 332

4 32

8

8

4

x

x

x

x

14

14

4144

1 2 0

1 2

1 2

1116

1716

x

x

x

x

x

Solving ExponentialsSolving Exponentials

Exponentiating:Exponentiating: eexx = 72 = 72 lnelnexx = ln72 = ln72 x = ln72 x = ln72 ≈ 4.277≈ 4.277

You should always check your answers by You should always check your answers by plugging them back in. Sometimes they don’t plugging them back in. Sometimes they don’t work because you can’t take the log of a negative work because you can’t take the log of a negative number.number.

Solving ExponentialsSolving Exponentials

3(23(2xx) = 42) = 42 22xx = 14 = 14 loglog2222xx = log = log221414 x = logx = log221414 x = log14/log2 x = log14/log2 ≈ 3.807≈ 3.807

x

x

x

AlternativeMethod :3(2 ) 422 14ln2 ln14xln2 ln14

ln14x 3.807ln2

Solving ExponentialsSolving Exponentials

4e4e2x2x – 3 = 2 – 3 = 2 4e4e2x2x = 5 = 5 ee2x2x = 5/4 = 5/4 lnelne2x2x = ln 5/4 = ln 5/4 2x = ln 5/42x = ln 5/4 x = ½ ln 5/4 x = ½ ln 5/4 ≈ 0.112≈ 0.112

Solving ExponentialsSolving Exponentials 2(32(32t-52t-5) – 4 = 11) – 4 = 11 2(32(32t-52t-5) = 15) = 15 (3(32t-52t-5)= 15/2)= 15/2 loglog33(3(32t-52t-5) = log) = log33 15/2 15/2 2t – 5 = log2t – 5 = log33 15/2 15/2 2t = 5 + log2t = 5 + log33 7.5 7.5 t = 5/2 + ½ logt = 5/2 + ½ log33 7.5 7.5 t t ≈ 3.417≈ 3.417

Solving ExponentialsSolving Exponentials

ee2x2x – 3e – 3exx + 2 = 0 + 2 = 0 No like terms—kinda look quadratic?No like terms—kinda look quadratic? (e(exx – 2)(e – 2)(exx – 1) = 0 – 1) = 0 Set each factor = 0 and solveSet each factor = 0 and solve (e(exx – 2) = 0 – 2) = 0 eexx = 2 = 2 lnelnexx = ln2 = ln2 x = ln2 x = ln2 ≈ 0.693≈ 0.693

(ex – 1) = 0

ex = 1

lnex = ln 1

x = 0

Solving LogarithmsSolving Logarithms

Exponentiating with the natural logExponentiating with the natural log lnx = 2lnx = 2 eelnxlnx = e = e22

x = ex = e22 ≈ 7.389≈ 7.389

Solving Logarithms Solving Logarithms

loglog33(5x - 1) = log(5x - 1) = log33(x + 7)(x + 7) 5x – 1 = x + 75x – 1 = x + 7 4x = 84x = 8 x = 2x = 2

Solving Logs – Last TimeSolving Logs – Last Time

5 + 2lnx = 45 + 2lnx = 4 2lnx = -12lnx = -1 lnx = - ½ lnx = - ½ eelnxlnx = e = e - ½- ½ x = e x = e - ½- ½

x x ≈ 0.607≈ 0.607

Interest Compounded ContinuouslyInterest Compounded Continuously

If interest is compounded “all the time” If interest is compounded “all the time” ((MUSTMUST use the word use the word continuouslycontinuously), we use ), we use the formulathe formula

where Pwhere P00 is the initial principle (initial is the initial principle (initial amount)amount)

0( ) rtP t P e

0( ) rtP t P e If you invest $1.00 at a 7% annual rate that If you invest $1.00 at a 7% annual rate that

is compounded continuously, how much will is compounded continuously, how much will you have in 4 years?you have in 4 years?

You will have a whopping $1.32 in 4 years!You will have a whopping $1.32 in 4 years!

(.07)(4)1* 1.3231e