Essential Question: Give examples of equations that can be solved using the properties of

exponents and logarithms.

An equation where the exponent is a variable is an exponential equation.

You solve exponential equations by converting them into logarithmic equations, and using the properties of logarithms to simplify.

As a rule: you need to get the base and exponent alone on one side of the equation first before converting to a log.

Example◦ Solve 73x = 20

log7 20 = 3x convert to log

change of base formula

divide both sides by 3

0.5132 = x use calculatorround to 4 decimal places

log 203

log7x

log 20

3log7x

Example (get base/exponent alone first)◦ Solve 5 - 3x = -40

-3x = -45 subtract 5 on both sides

3x = 45 divide both sides by -1

log3 45 = x convert to log

change of base formula

3.4650 = x use calculatorround to 4 decimal places

log 45

log3x

Your Turn◦ Solve 3x = 4

◦ Solve 62x = 21

◦ Solve 3x+4 = 101

1.2619

0.8496

0.2009

Assignment◦ Page 464◦ Problems 1 – 19 (odds)

◦ Show your work, and round your answers to 4 decimal places

◦ Ignore the directions about solving by graphing and using a table.

Essential Question: Give examples of equations that can be solved using the properties of

exponents and logarithms.

An equation that includes a logarithmic expression, such as log3 15 = log2 x is called a logarithmic equation.

You solve logarithmic equations by using the properties of logarithms to simplify and then converting them into exponential equations.

As a rule: you need to get the logs on one side of the equation and combined into only one log before converting to an exponential equation.

As another rule: If there is no base on a logarithmic problem, we assume the base is 10

Example◦ Solve log (3x + 1) = 5 Only one log? Check

log10 (3x + 1) = 5 Assume base 10

105 = 3x + 1 Convert to exponential form

100,000 = 3x + 1 Simplify left side99,999 = 3x Subtract 1 from both

sides33,333 = x Divide both sides by 3

Example (combining logs first)◦ Solve 2 log x – log 3 = 2

log x2 – log 3 = 2 Power rule

Quotient RuleOnly one log? Check

Assume base 10

102 = Convert to exponential form

100 = Simplify left side300 = x2 Multiply both sides by 317.3205 = x Square root both sides

2

log 23

x

2

10log 23

x2

3

x

2

3

x

Your Turn◦ Solve log (7 – 2x) = -1

◦ Solve log (2x – 2) = 4

◦ Solve 3 log x – log 2 = 5

◦ Solve log 6 – log 3x = -2

3.45

5001

58.4804

200

Assignment◦ Page 464 - 465◦ Problems 33 – 47 (odds)

◦ Show your work, and round your answers to 4 decimal places (if necessary)

◦ Ignore the directions about solving by graphing and using a table.