Solving Exponential & Logarithmic Equations Lesson Notes

Two forms

1. exponential

2. logaerithmic

It helps to think about logarithms as exponents. Rewriting can help you figure it out.

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Rewriting Logarithms

Multiple Choice

Rewrite log₂8 = 3 in exponential form

• 2⁸ = 3

• 2³ = 8

• 3² = 8

• 8³ = 2

Multiple Choice

Write in exponential form.
log₂32 = 5

• 2^-5 = 32

• 2³² = 5

• 2⁵ = 32

• 32⁵ = 2

Multiple Choice

Rewrite 3⁴ = 81 in logarithmic form.

• log₃4 = 81

• log₈₁3 = 4

• log₃81 = 4

• log₄81 = 3

Multiple Choice

Rewrite log₃81 = 4 exponentially

• 3⁴ = 81

• 4³ = 81

• 81³=4

Change of Base

Wait... What was that log₇46??

We need a way to go back and forth between base 10 and other bases.

This is called the CHANGE OF BASE formula!

Multiple Choice

Which property of logarithms is demonstrated below: 

$$\log_920\ =\ \frac{\log20}{\log9}$$  

• Product property

• Quotient property

• Power property

• Change of Base Property

​PROPERTIES OF LOGARITHMS

Multiple Choice

Write log_b(xy) as two logs

• log_bx+log_by

• log_bx-log_by

• log_bx*log_by

• log_bx/log_by

Multiple Choice

Write log_b(x/y) as two logs

• log_bx-log_by

• log_bx+log_by

• log_bx*log_by

• log_bx/log_by

Multiple Choice

Rewrite log_b(xⁿ)

• nlog_bx

• (log_bx)ⁿ

• xⁿlog_bx

• log_b(xⁿ)

Multiple Choice

Use the properties of logarithms to retwrite

• A

• B

• C

• D

Multiple Choice

Rewrite as a single logarithm:

$$\log3\ +\ \log7$$  

• log 10

• log 21

• log 3/7

• log 3/log 7

Multiple Choice

Rewrite as a single logarithm:

$$\log_260\ -\ \log_210$$  

• $$\log_26$$  

• $$\log_250$$  

• $$\frac{\log_260}{\log_210}$$  

• $$\log_270$$  

Multiple Choice

Use the properties of logarithms to rewrite as the sum of two logarithms:

$$\log55$$  

• log 40 + log 15

• log 50 + log 5

• log 11 + log 5

Solving Exponential Equations

• Second Method: When there is an exponent on only ONE side.

Multiple Choice

Solve for x. $$\log_7x=4$$  

• 16384

• 2401

• 11

• 1.75

Multiple Choice

Solve this one!

$$3\cdot4^{x-1}=24$$  

• $$x=1$$  

• $$x=\frac{\log8}{\log4}+1$$  

• $$x=\frac{\log21}{\log4}+1$$  

• $$x=1.5$$  

This is how the last problem should be solved!!

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

Expand: log(4x)

• log4-logx

• log4+logx

• 4logx

• xlog4

Multiple Choice

Rewrite as a single logarithm:

$$\log3\ +\ \log7$$  

• log 10

• log 21

• log 3/7

• log 3/log 7

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

Expand:   $$\log\left(\frac{x}{y}\right)$$  

• logx+logy

• xlogy

• log(x-y)

• logx-logy

Multiple Choice

Expand $$\log_6\left(\frac{y}{36}\right)$$  

• $$\log_6y+\log_636$$  

• $$\log_6y-\log_636$$  

• $$\log_636+\log_6y$$  

• $$\log_636-\log_6y$$  

Power Property

Power Property

Examples

Additional rules

Examples

Solving Exponential Equations

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1.Rewrite both sides of the equation with the same base.

2.Set the exponents equal to one another.

3.Solve for x.​

5^x = 625​

5^x = 5⁴

x = 4​

Solving Exponential Equations

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1.Rewrite both sides of the equation with the same base.

2.Set the exponents equal to one another.

3.Solve for x.​

3^2x = 81​

3^2x = 3⁴

2x = 4​

x = 2​

Solving Exponential Equations

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1.Rewrite both sides of the equation with the same base.

2.Set the exponents equal to one another.

3.Solve for x.​

3^{2x + 1} = 3^{1 - x}

2x + 1 = 1 - x

2x + x = 1 - 1​

3x = 0

x = 0​

Solving Exponential Equations

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1.Rewrite both sides of the equation with the same base.

2.Set the exponents equal to one another.

3.Solve for x.​

(⅓)^x = 81

3^-x = 3⁴

-x = 4

x = -4​

Solving Exponential Equations

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1.Rewrite both sides of the equation with the same base.

2.Set the exponents equal to one another.

3.Solve for x.​

9^{2x - 1} = 3^8x

(3²)^{2x - 1} = 3^8x

3^{4x - 2} = 3^8x

4x - 2 = 8x

-2 = 8x + 4x

-2 = 12x​​

x = ​-½

Exponential to Logarithmic

​5⁰ = 1

log₅1 = 0​

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2⁵ = 32​

log₂32 = 5​

​3^-2 = 1/9

log₃1/9 = -2​

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4^x = 16​

log₄16 = x​

Logarithmic to Exponential

​log₄64 = 3

4³ = 64​

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log₁₄₃20,449 = 2

143² = 20,449​

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​log_bm = n

bⁿ = m

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log₂⅛ = -3

2^-3​ = ⅛

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