Chapter 3:Exponential & Logarithmic Functions

Exponential Rules

100

The function denoted by f(x)=a^x Identify the base

 base a?

200

rewrite the logarithm to exponential form.

log₃ 81 = 4

3⁴ =81

300

The exponential rule that Evaluate 

(x³)²

power of a power

400

Simplify the following 

(18x³zy⁵ . 3x³wy) / (27xzy³)

54 x⁶zy⁶ / (27xzy³)

500

if you let the number of compoundings n increase without bound, the process approaches ...

what is continuous compounding?

Logarithmic rules

100

Rewrite the exponential equation to log form.

12² = 144

 log₁₂ 144 = 2

200

the logarithmic function with base 10

what is the common logarithmic function?

300

what is the base of a natural log?

e

400

Identify the property of logarithm shown in the example below.

log x - log y 

Quotient property

500

Rewrite the equation below in logarithmic form 

25⁰ = 1

log₂₅ 1 = 0

Properties of Logs

100

Condense the following, using the properties of logarithms

       6log₃ u + 6log₃ v

    log 3 (u⁶ v⁶)

200

condense the following using properties of logarithms:                   2 log 7

                     3

         log (7²)^1/3

or log of the cube root of 7²

300

Expand the following using properties of logarithms:               log ( t/c³)⁷

7 log t - 21 log c

400

condense the following using properties of logarithms: log_auⁿ

what is nlog_au?

500

expand the following using properties of logarithms:

       log₅ (x. y². w)

log₅ x + 2log₅ y + log₅ w

Solving logs and exponential functions

100

Solve the equation below

log (4k − 5) = log (2k − 1)

 log (4k − 5) = log (2k − 1)

              4k-5 = 2k -1

             4k-2k= -1+5

                   2k = 4

                     k = 2

200

Solve the equation below

3 ^{1 − 2x} = 243

3 ^{1 − 2x} = 243

3 ^{1 − 2x} =3⁵

1 - 2x = 5

1 - 5 = 2x

-4 = 2x

-2 = x

300

Solve the following 

4^5m-3 = 4¹⁰

4^5m-3 = 4¹⁰

5m -3 = 10

5m =13

m = 13/5    or 2.6

400

Solve the equation. Round your answers to the nearest ten-thousandth.

6e ^{5x − 6} − 4 = 50

       6e ^{5x − 6} − 4 = 50

        6e ^{5x − 6}= 54

         e ^{5x − 6} = 9

        5x - 6= ln(9)

         x = (2.1972+6) / 5

         x = 1.6394

500

Using a calculator to approximate each of the following.

    ln(-30)

Undefined

Application questions

100

The total  gallon of Orange juice each person consumes yearly has decreased 4.1% each year. In 1981 each person drank 16.5 gallons of orange Juice on average. What was the approximate amount consumed in 2010

  y = a.b^x

  y = 16.5 ( 1 - 0.041)²⁹

  y = 4.9

4.9 gallons of juice consumed per person

200

You buy a car for $7500 that depreciated at a rate of 15% a year. 

a) How much is the car worth after 4 years?

b) When will the car be worth  $2500 or less

  y = a.b^x

  y = $7500(1- 0.15)⁴

  y =$3915. 05

  $2500 = $7500 (1 - 0.15)^t 

   log (2500/7500) = T log(0.85)  

    T = 7 years

300

Mary invested $5000 into an account that earns 8% interest compounded continuously. After 12 years she has approximately $30000 in the account. What is the rate at which it was compounded?

A = Pe^rt

$30000 = $5000 .e^r(12)

ln (30000/5000) = 12r

1.7918 / 12 = r

R = 0.1493  

400

When Jacob was born $18000 was invested in his trust fund at a rate of 6.25% interest, compounded continuously.

a) What was the balance in Jacob's account at age 15?

b) How old will Jacob be when his investment reaches $75000

   A =Pe^rt

    A =18000.e ^{0.0625 (15)}

   A = $45964. 61

b) $75000 = $18000 .e ^{0.0625 (t)}

      ln ($75000/$18000) = 0.0625 t

      ln (1.4271) / 0.0625 = t

         T = 23 years (approximately)

500

A population of 3000 people doubles in size every 10 years. What is the population size after 50 years?

   y =a.b^x

   y = 3000(2)⁵

   y = 96000

96000 people after 50 years.