Exponential and Logarithmic Equations

Solving Simple Equations

Solving Exponential Equations

Solving An Exponential Equation Of Quadratic Type

Solving Logarithmic Equations

Key Vocabulary and Properties

100

2^x=32

Solve for x

2^x=32

1. Rewrite with common base 2^x=2⁵

2. The bases cancel out x=5

100

e^x+5=60

Solve for x

1. Take 5 from each side e^x=55

2. Take natural log of each side ln e^x=ln 55

3. Impliment the Inverse Property x=ln 55/x=4.007

100

e^2x-4e^x-5=0

Solve for x

1. Rewrite the equation (e^x)²-4e^x-5

2. Factor (e^x-5)(e^x+1)

3. Rewrite e^x-5=0

4. Multiply by ln and solve for x

x=ln5

x=1.609

100

lnx= -3

Approximate the result to 3 decimal places

1. Do the circle thingy

3^-3=x

2. Solve

e^-3= -0.223

100

What is the One-To-One Property and how is it used to solve a logarithmic equation?

a^x=a^y

The One-To-One Property is if Y=X then it is One-To-One

You apply the Property to a logarithmic equation by if the base of a logarithm on either side is the same then they cancel out.

200

5-e^x=0

Solve for x

1. Move e on other side 5=e^x

2. Use natural log to cancel out base of e ln 5=ln e^x

3. The ln cancels out e. Rewrite ln 5=x

200

2(5^x)=32

Solve for x

1. Divide each side by 2 5^x=16

2. Take log₅to isolate x log₅5^x=log₅16

3 . Rewrite

x=log₅16

4. Solve using the change of base property log 16 over log 5 (log16)/(log5)

5. Solve for x x=1.722

200

e^2x-5e^x+6=0

Solve for x

1. Rewrite using substitution

(e^x)²-5e^x+6=0

2. Factor

(e^x-3)(e^x-2)=0

3. Rewrite

e^x-3=0

4. Take ln of each side and solve for x

x=ln3

x=1.098

200

log₄x-log₄(x-1)=1/2

Approximate the result to three decimal places, if necessary

1. Rewrite the equation using the Quotient Property

log₄(x/x-1)=1/2

2. Rewrite

(x/x-1)=4^1/2

3. Write in exponential form

(x/x-1)=(2²)^1/2

4. Simplify

(x/x-1)=2

5. Multiply by x-1

x=2(x-1)

6. Distribute

x=2x-2

7. Solve for x

x=2

200

What are the strategies for solving exponential and logarithmic equations?

1. Rewrite fthe origional equation in a form that allows the use of the One-To-One Properties of exponential or logarithmic functions.

2. Rewrite an exponential equation in logarithmic form and apply the Inverse Property of logarithmic functions.

3. Rewrite a logarithmic equation in exponential form and apply the Inverse Property of exponential functions.

300

log₆x=3

Solve for x

1. Use the Circle method thingy and rewrite as an exponential equation 6³=x

2. Solve x=216

300

e^x-7=23

Solve for x

1. Subtract 7 from each side e^x=30

2. Take natural log of each side lne^x=ln30

3. Rewrite to isolate x x=ln30

4. Solve x=3.401

300

e^2x-7e^x-12=0

Solve for x

1. Rewrite using substitution

(e^x)²-7e^x-12=0

2. Factor

(e^x-3)(e^x-9)

3. Rewrite

e^x-3=0

4. Take ln of each side

x=ln3

x=1.098

300

log₂(x+2)+log₂(3)=log₂(27)

Solve for x

1. Reduce log₂(27)

log₂(x+2)+log₂(3)=log₂(3³)

2. Drop exponent

log₂(x+2)+log₂(3)=3log₂(3)

3. Move constant to the right side of the equation.

log₂(x+2)=3log₂(3)-log₂(3)

4. Combine like terms

log₂(x+2)=2log₂(3)

5. Use the Power Property and lift exponent

log₂(x+2)=log₂(3²)

6. Bases cancel out

x+2=3²

7. Solve for x

x=7

300

What the Inverse Property

The Inverse Property states that a number added to it's opposite will equal zero.

400

What is the Quotient Rule?

The log of a quotient is equal to the difference of the logs of the numerator and denominator.

500

What is the Product Rule?

The log of a product is equal to the sum of the log of the first base and the log of the second base.