Exponential and Logarithmic Equations

LESSON 3–4

A. Solve 4x + 2 = 16x – 3.

4x + 2= 16x – 3 Original equation4x + 2= (42)x – 3 42 = 164x + 2 = 42x – 6 Power of a Powerx + 2= 2x – 6 One-to-One Property2 = x – 6 Subtract x from each side.8 = x Add 6 to each side. Answer: 8

Solve Exponential Equations Using One-to-One Property

B. Solve .

Original equation

Power of a Power

One-to-One Property.

A. 1

B.

C. 2

D. –2

Solve 25x + 2 = 54x.

Solve Logarithmic Equations Using One-to-One Property

A. Solve 2 ln x = 18. Round to the nearest hundredth.

Method 1 Use exponentiation. 2 ln x

= 18 Original equationln x

= 9 Divide each side by 2.eln x

= e9 Exponentiate each side.x

= e9 Inverse Propertyx

≈ 8103.08 Use a calculator.

Method 2 Write in exponential form.2 ln x= 18 Original equationln x= 9 Divide each side by 2.x= e9 Write in exponential form.x≈ 8103.08 Use a calculator.

Answer: 8103.08

Solve Logarithmic Equations Using One-to-One Property

B. Solve 7 – 3 log 10x = 13.

7 – 3 log 10x = 13Original equation–3 log 10x= 6Subtract 7 from each side.log 10x = –2 Divide each side by –3.10–2 =10x Write in exponential form.10–3 = x Divide each side by 10.

= x = 10–

3.

Answer:

Solve 2 log2x 3 = 18.

A. 81

B. 27

C. 9

D. 8

A. Solve log2 5 = log2 10 – log2 (x – 4). log25= log210 – log2(x – 4)Original equationlog25 = Quotient Property

5 = One-to-One Property

5x – 20= 10Multiply each side by x – 4.55x = 30 Add 20 to each side.x = 6 Divide each side by 5. Answer: 6

Solve Exponential Equations Using One-to-One Property

B. Solve log5 (x2 + x) = log5 20.

log5(x2 + x) = log520

Original equationx2 + x = 20 One-to-One Propertyx2 + x – 20= 0Subtract 20 from each side.(x – 4)(x + 5) = 0Factor x

2 + x – 20 into linear factors.x = –5 or 4 Solve for x. Check this solution.

Answer: –5, 4

Solve log315 = log3x + log3(x – 2).

A. 5

B. –3

C. –3, 5

D. no solution

Solve Exponential EquationsA. Solve 3x = 7. Round to the nearest hundredth.

3x = 7 Original equationlog 3x= log 7Take the common logarithm of each side.x log 3= log 7Power Propertyx= or about 1.77Divide each side by log 3 and use a calculator.Answer: 1.77

B. Solve e2x + 1 = 8. Round to the nearest hundredth. e2x + 1 = 8 Original equationln e2x + 1= ln 8Take the natural logarithm of each side.2x + 1= ln 8Inverse Propertyx= or about 0.54Solve for x and use a calculator.Answer: 0.54

Solve 4x = 9. Round to the nearest hundredth.

A. 0.63

B. 1.58

C. 2.25

D. 0.44

Solve in Logarithmic Terms

Solve 36x – 3 = 24 – 4x. Round to the nearest hundredth.

36x – 3 = 24 – 4x

Original equationln 36x – 3= ln 24 – 4x Take the natural logarithm of each side.(6x – 3) ln 3= (4 – 4x) ln 2Power Property6x ln 3 – 3 ln3= 4 ln 2 – 4x ln 2Distributive Property6x ln 3 + 4x ln 2= 4 ln 2 + 3 ln3Isolate the variable on the left side of the equation.

x(6 ln 3 + 4 ln 2) = 4 ln 2 + 3 ln3Distributive Property

(could just divide here… hard to type in)

Solve in Logarithmic Terms

x(ln 36 + ln 24)= ln 24 + ln 33Power Propertyx ln [36(24)]= ln [24(33)]Product Propertyx ln 11,664= ln 43236(24) = 11,664 and 24(33) = 432x= Divide each side by ln 11,664.

Answer: 0.65

x≈ 0.65Use a calculator.

Solve 4x + 2 = 32 – x. Round to the nearest hundredth.

A. 1.29

B. 1.08

C. 0.68

D. –0.23

Solve e2x – ex – 2 = 0.e2x – ex – 2 = 0

Original equationu2 – u – 2 = 0

Write in quadratic form by letting u = ex.(u – 2)(u + 1)= 0

Factor.u = 2 or u = –1Zero Product Propertyex = 2 ex

= –1 Replace u with ex.

Solve Exponential Equations in Quadratic Form

ln ex = ln 2 ln ex = ln (–1) Take the natural logarithm of each side.

x = ln 2 x = ln (–1) Inverse Property or about 0.69

The only solution is x = ln 2 because ln (–1) is extraneous.Answer: 0.69

Solve e2x + ex – 12 = 0.

A. ln 3

B. ln 3, ln 4

C. ln 4

D. ln 3, ln (–4)

Solve Logarithmic Equations

Solve log x + log (x – 3) = log 28.

log x + log (x – 3)= log 28Original equation

log x(x – 3)= log 28Product Property

log (x 2 – 3x)= log 28

Simplify.x

2 – 3x = 28 One-to-One Propertyx

2 – 3x – 28 = 0 Subtract 28 from each side.(x – 7)(x + 4) = 0 Factor.x = 7 or x = – 4 Zero Product Property

The only solution is x = 7 because –4 is an extraneous solution.

Answer: 7

Solve log (3x – 4) = 1 + log (2x + 3).

log (3x – 4)

= 1 + log (2x + 3)

Check for Extraneous Solutions

log (3x – 4) – log (2x + 3)

= 1= 1

= log 101

= log 10

= 103x – 4

= 10(2x + 3)

3x – 4

= 20x + 30

–17x

= 34

x

= –2

But wait… must check for extraneous solutions

Answer: no solution

Check for Extraneous Solutions

Checklog (3x – 4) = 1 + log (2x + 3)

log (3(–2) – 4) = 1 + log (2(–2) + 3)log (–10) = 1 = log (–1)

Since neither log (–10) nor log (–1) is defined, x = –2 is an extraneous solution.

Model Exponential Growth

A. CELL PHONES This table shows the number of cell phones a new store sold in March and August of the same year. If the number of phones sold per month is increasing at an exponential rate, identify the continuous rate of growth. Then write the exponential equation to model this situation.

Model Exponential Growth

Let N(t) represent the number of cell phones sold at the end of t months and assume continuous growth. Then the initial number N0 is 88 cell phones sold and the number of cell phones sold N after a time of 5 months, the number of months from March to August, is 177. Use this information to find the continuous growth rate k.

N (t) = N0ekt Exponential

Growth Formula

177 = 88e5k N(5) = 177, N0 = 88, and t = 5= e5k Divide each side by 88.

Answer: 13.98%; N (t) = 88e0.1398t

Model Exponential Growth

= ln e5k Take the natural logarithm of each side.

= 5k Inverse Property

= k Divide each side by 5

0.1398 ≈ k Use a calculator.

Model Exponential Growth

B. CELL PHONES Use your model to predict the number of months it will take for the store to sell 500 phones in one month.

N (t) = 88e0.1398t

N (t) = 88e0.1398t

500 = 88e0.1398t

= ln e0.1398t

= e0.1398t

= 0.1398t

= t

12.4 ≈ t

According to this model, the store will sell 500 phones in a month in about 12.4 months.Answer: 12.4 months