Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

[email protected] • MTH55_Lec-64_sec_9-5a_Exponential_Eqns.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§9.5a§9.5aExponential EqnsExponential Eqns



Review §Review §

Any QUESTIONS About• §9.4 → Logarithm Change-of-Base

Any QUESTIONS About HomeWork• §9.4 → HW-47

9.4 MTH 55



Summary of Log RulesSummary of Log Rules

For any positive numbers M, N, and a with a ≠ 1

log log log ;a a aM

M NN

log log ;pa aM p M

log .ka a k

log ( ) log log ;a a aMN M N



Typical Log-ConfusionTypical Log-Confusion

BewareBeware that Logs do NOT behave Algebraically. In General:

loglog ,

loga

aa

MM

N N

log ( ) (log )(log ),a a aMN M N

log ( ) log log ,a a aM N M N

log ( ) log log .a a aM N M N



Solving Exponential EquationsSolving Exponential Equations

Equations with variables in exponents, such as 3x = 5 and 73x = 90 are called EXPONENTIAL EQUATIONS

Certain exponential equations can be solved by using the principle of exponential equality



Principle of Exponential EqualityPrinciple of Exponential Equality

For any real number b, with b ≠ −1, 0, or 1, then

bx = by is equivalent to x = y

That is, Powers of the same base are equal if and only if the exponents are equal



Example Example Exponential Equality Exponential Equality

Solve for x: 5x = 125 SOLUTION Note that 125 = 53. Thus we can write

each side as a power of the same base:

5x = 53

Since the base is the same, 5, the exponents must be equal. Thus, x must be 3. The solution is 3.



Example Example Exponential Equality Exponential Equality

Solve each Exponential Equationa. 25x 125 b. 9x 3x1

SOLUTION

a. 52 x53

52 x 53

2x 3

x 3

2

b. 32 x3x1

32 x 3x1

2x x 1

2x x 1

x 1



Principle of Logarithmic EqualityPrinciple of Logarithmic Equality

For any logarithmic base a, and for x, y > 0,

x = y is equivalent to logax = logay

That is, two expressions are equal if and only if the logarithms of those expressions are equal



Example Example Logarithmic Equality Logarithmic Equality

Solve for x: 3x+1 = 43 SOLUTION

3 x +1 = 43

log 3 x +1 = log 43

(x +1)log 3 = log 43

x +1 = log 43/log 3

x = (log 43/log 3) – 1 2.4236.x

Principle of logarithmic equality

Power rule for logs

The solution is (log 43/log 3) − 1, or approximately 2.4236.



Example Example Logarithmic Equality Logarithmic Equality

Solve for t: e1.32t = 2000 SOLUTION

5.7583.t

Note that we use the natural logarithm

Logarithmic and exponential functions are inverses of each other

e1.32t = 2000

ln e1.32t = ln 2000

1.32t = ln 2000

t = (ln 2000)/1.32



To Solve an Equation of the To Solve an Equation of the Form Form aatt = = bb for for tt

1. Take the logarithm (either natural or common) of both sides.

2. Use the power rule for exponents so that the variable is no longer written as an exponent.

3. Divide both sides by the coefficient of the variable to isolate the variable.

4. If appropriate, use a calculator to find an approximate solution in decimal form.



Example Example Solve by Taking Logs Solve by Taking Logs

Solve each equation and approximate the results to three decimal places.

a. 2x 15 b. 52x 2 17

SOLUTION a. 2x 15

ln 2x ln15

x ln 2 ln15

x ln15

ln 23.907



Example Example Solve by Taking Logs Solve by Taking Logs

SOLUTION

b. 52x 3 17

2x 3 17

5

ln 2x 3 ln17

5

x 3 ln 2 ln17

5

x 3 ln

175

ln 2

x ln

175

ln 2 3

x 4.766

b. 52x 2 17



Example Example Different Bases Different Bases

Solve the equation 52x−3 = 3x+1 and approximate the answer to 3 decimals

SOLUTION ln 52 x 3 ln 3x1

2x 3 ln 5 x 1 ln 3

2x ln 5 3ln 5 x ln 3 ln 3

2x ln 5 x ln 3 ln 3 3ln 5

x 2 ln 5 ln 3 ln 3 3ln 5

x ln 3 3ln 5

2 ln 5 ln 32.795

Take ln of both sides



Example Example Eqn Quadratic in Form Eqn Quadratic in Form

Solve for x: 3x − 8∙3−x = 2. SOLUTION 3x 3x 83 x 2 3x

32 x 830 23x

32 x 8 23x

32 x 23x 8 0

This equation is quadratic in form. Let y = 3x then y2 = (3x)2 = 32x. Then,




Solncont.

32 x 23x 8 0

y2 2y 8 0

y 2 y 4 0

y 2 0 or y 4 0

y 2 or y 4

3x 2 or 3x 4 But 3x = −2 is not possible because

3x > 0 for all numbers x. So, solve 3x = 4 to find the solution




Solncont.

3x 4

ln 3x ln 4

x ln 3 ln 4

x ln 4

ln 3x 1.262



Example Example Population Growth Population Growth

The following table shows the approximate population and annual growth rate of the United States of America and Pakistan in 2005

CountryPopulatio

n

Annual Population

Growth Rate

USA 295 million 1.0%

Pakistan 162 million 3.1%




Use the population model P = P0(1 + r)t and the information in the table, and assume that the growth rate for each country stays the same.

In this model, • P0 is the initial population,

• r is the annual growth rate as a decimal

• t is the time in years since 2005




Use P = P0(1 + r)t and the data table:

a. to estimate the population of each country in 2015.

b. If the current growth rate continues, in what year will the population of the United States be 350 million?

c. If the current growth rate continues, in what year will the population of Pakistan be the same as the population of the United States?




SOLUTION: Use model P = P0(1 + r)t

a. US population in 2005 is P0 = 295. The year 2015 is 10 years from 2005.

P 295 1 0.01 10 325.86 million

Pakistan in 2005 is P0 = 162

P 162 1 0.31 10 219.84 million




SOLUTION b.: Solve for t to find when the United States population will be 350.

350 295 1 0.01 t

350

295 1.01 t

ln350

295

ln 1.01 t

ln350

295

t ln 1.01

t ln

350295

ln 1.01 17.18

Some time in yr 2022 (2005 + 17.18) the USA population will be 350 Million




SOLUTION c.: Solve for t to find when the population will be the same in both countries. 295 1 0.01 t 162 1 0.031 t

295 1.01 t 162 1.031 t

295

162

1.031

1.01

t

ln295

162

ln

1.031

1.01

t




Soln c.cont. ln

295

162

t ln

1.031

1.01

t ln

295162

ln1.0311.01

29.13

Some time year 2034 (2005 + 29.13) the two populations will be the same.



WhiteBoard WorkWhiteBoard Work

Problems From §9.5 Exercise Set• 16, 20, 32, 34, 36, 40

logistic difference equation by Belgian ScientistPierre Francois Verhulst



All Done for TodayAll Done for Today

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Calculator



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

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