Essential Question: Give examples of equations that can be solved using the properties of exponents and logarithms.
Jan 09, 2016
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.