11/7 Review on Exponential and Logs

​Reviewing Exponentials

& Logs

By C. Epley

Converting Logs and Exponentials

Converting examples:

Multiple Choice

Convert log(x)=5 to exponential form

• 1⁵=x
• 10^x=5
• x⁵=10
• 10⁵=x

Multiple Choice

Convert 6^(x+2)=216 to logarithmic form

• log₂₁₆(x+2)=6
• log₆(x+2)=216
• log₆(216)=x+2
• log_(x+2)(216)=6

Multiple Choice

Convert log₈(x+2)=3 to exponential form.

• 8^(x+2)=3
• 3^(x+2)=8
• 8³=x+2
• 3⁸=x+2

Multiple Choice

Convert 2^x+6=10 to logarithmic form

• log₄(x)=2

• log₂(4)=x

• log₂(10)=6

• log₂(x)=4

Exponent Rules

Since logs are exponents they follow many of the rules of exponents

There are 3 Log Properties

Product Property

Echoes the multiplication rule of exponents

A product of 2 expressions within a log can be expanded into a sum of those expressions

Multiple Choice

 Choose the correct expanded common logarithm: $$\log(4x)$$

• $$\log4-\log x$$  

• $$\log4+\log x$$  

• $$4\log x$$  

• $$x\log4$$  

Quotient Property

Echoes the Division Rule of Exponents

When 2 expressions within a log are divided they can be expanded into a difference of those expressions

Quotient Examples

Notice when condensing we write only ONE log term!

Multiple Choice

Choose the correct expanded common logarithm: $$\log\left(\frac{x}{y}\right)$$  

• $$\log x+\log y$$  

• $$x\log y$$  

• $$\log(x-y)$$  

• $$\log x-\log y$$  

Power Property

Power Property

In the example, the exponent of 3 can be brought down in front of the expression. We will use this property frequently when solving expon. equations.

Condensing/Expanding when more than 1 property is needed:

• Follow the order of operations

• When condensing, always do the Power Property first

Multiple Choice

log (x²)

• 2 log (x)

• log (x)+log (2)

• log (x) * log (y)

• x log (2)

Multiple Choice

Expand completely: log(2x⁵) 

• log 2 + 5log x

• 5log 2 + 5log x

• 5log 2 + log x

• log 10 + log x

Multiple Choice

Choose the correctly condensed common logarithm $$\log x+\log y+3\log z$$  

• $$\log\left(xyz^3\right)$$  

• $$\log\left(xyz\right)^3\ \ \ \$$  

• $$\log\left(3xyz\right)\ \ \ \ \$$  

• $$3\log\left(xyz\right)$$  

Multiple Choice

Expand

• 6log₈v-2log₈u

• 6log₈u-2log₈v

• 3log₈u-2log₈v

• 6log₈u+2log₈v

Multiple Choice

Condense into one log:   $$2\log_3\left(x\right)+\log_3\left(y\right)$$   Remember to use the power property first!

• $$\log_3\left(2xy\right)\$$

• $$\log_3\left(x^2y\right)$$

• $$\log_3\left(xy\right)^2$$

• $$2\log_3\left(xy\right)$$

Multiple Choice

Condense into a single log expression.

Multiple Choice

Expand:    $$\log\left(\frac{m^3}{n}\right)$$  

• logm-log3-logn

• 3logm+logn

• 3logm-logn

• 3log(m-n)

Solving Log Equations

Things to keep in mind:

-Are the bases of the logs the same?

- Is there any property we can use to simplify the equation?

A logarithmic equation is an equation containing a variable in a logarithmic expression.

2 Methods to Solve Log Equations

Multiple Choice

What is the first step you would take to solve this log equation?

$$\log_32x=4$$  

• Subtract 4

• Convert to exponential

• Subtract  $$\log_32x=4$$  

• Divide by  $$2x$$  

Multiple Choice

Solve:

log₃(x - 5) = 2

• 25

• 5

• 130

• 14

Multiple Choice

Solve for x:

log₈(4x+4)=2

• 15

• 12

• 10

• 3

Fill in the Blank

$$\log_4\left(x+1\right)=2$$  

x=_____

Fill in the Blank

$$\log\left(x-3\right)=2$$  

x=_____

Multiple Choice

What is the first step you would take to solve this log equation?

$$\ln\left(2x\right)=\ln\left(5x+8\right)$$  

• Subtract one of the logs

• Set  $$2x=5x+8$$  

• Divide one of the logs

• Added 'e' as a base

Multiple Choice

Solve.

• 9

• 5

• 6

• -5

Multiple Choice

Solve.

• 32

• 40

• 36

• 16

Draw

Solve the equation. Be sure to show all work.

2 log n = log 256

An exponential equation is an equation that includes a variable in an exponent.

Powers of the Same Base Method

• Express each side as a power of the same base.

• Set the powers equal to each other and solve for your variable.

Multiple Choice

8^3x+3 = 8⁶

• x=28
• x=3
• x=1
• x=12

Multiple Choice

Sovle for x:
5^{-3x - 1} = 25

• x = -1
• x = -4
• x = -3
• x = 1

Multiple Choice

Solve.

• 3.5

• 3

• 4

• 4.5

Multiple Choice

Solve.

• 2

• 4

• 8

• 16

Use Logs to Solve Exponentials

• Isolate the exponential expression

• Take the log of each side

• Simplify

• Solve for your variable

Multiple Choice

Solve.

• 5.93

• 0.17

• 1.48

• -1.48

Solving With Logs of BOTH Sides of =

• Condense logs, if possible (Addition to Multiplication, Subtraction to Division)

• Logs cancel each other out

• Solve the remaining equation

Multiple Choice

Solve log₂(2x - 3) = log₂(4x + 5)

• 4

• -4

• -.33

• .33

Multiple Choice

Solve: $$\log_7\left(28+3r\right)=\log_7\left(10r\right)$$  

• 4

• 7

• 20

• 28

Multiple Choice

Solve: log₃ (5) + log₃(x) = log₃(2) + log₃(x + 9)

• 5

• 11

• 6

• 3

Solving with Logs on ONE Side of =

• Condense logs, if possible (Addition to Multiplication, Subtraction to Division)

• Isolate the log term (Add/Subtract, Divide by Coefficient)

• Change to Exponential Form

• Solve the Remaining Equation

Multiple Choice

log₆(2x + 3) = 3

• x = 106.5

• x = 100

• x = 16

• x = 50

Multiple Choice

log₃ (x-3) + 10 = 14

• 5

• 13

• 84

• -20

Multiple Choice

log₂(2) + log₂(8x) = 6

• x = 3
• x = 2
• x = 6
• x = 4

Solving Exponentials for Missing Exponents

• Isolate the exponential term (Add/Subtract, Divide by Coefficient)

• Change to logarithmic form

• Solve remaining equation (Change of Base Formula)

Multiple Choice

3^{2x – 6}  = 81

• x = log 4
• x = 5
• x = 4
• x = -1

Multiple Choice

7ⁿ⁺¹⁰- 8 = 6

• -7.374
• -8.643
• -7.360
• -8.853