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Solve Exponential Equations
Today’s Objectives:• Review how to solve exponential
equations that do not require logarithms.
• Solve exponential equations that require logarithms.
Type 1: Solve exponential equations without logarithms (review 7.2)
If bx = by, then x=y for b≠1.
If two powers with the same base are equal, then their exponents are equal.
43x = 8x+1Steps:1. Rewrite both sides of the equation with the same base
2. Set the exponents equal to one another.
3. Solve for x
4. Check your solution
Type 1: Solve exponential equations without logarithms
812x= 27x-1/9xSteps:1. Rewrite both sides of the equation with the same base
2. Set the exponents equal to one another.
3. Solve for x
4. Check your solution
Type 1: Solve exponential equations without logarithms (review 7.2)
Your turn! Complete these on your whiteboard/w partner
Evaluate the following logarithms:
What if I can’t write both sides using the same base????We will rewrite the equation using logarithms
102x-3+4 = 21Steps:1.Isolate the base with the variable exponent on one side of the equal sign.2.Take the log with the same base of each side3.Solve for x.4.Use change of base property, if needed.5.Check for extraneous solutions.
Type 2: Solve exponential equations with logarithms
½ex+2 + 3 = 25Steps:1.Isolate the base with the variable exponent on one side of the equal sign.2.Take the log with the same base of each side3.Solve for x.4.Use change of base property, if needed.5.Check for extraneous solutions.
Type 2: Solve exponential equations with logarithms
Your turn! Complete these on your whiteboard/w partner
33(2 ) 1 10
8x
–3e-x – 4 = –13
Suppose you deposit $5000 into a bank account that receives 2.4% interest monthly. How many years will it take for your account to reach $10,000?
Now, we can solve this algebraically!
Your turn!Complete these on the whiteboards with your
partner!
Suppose you deposit $2,000 into a bank account where the interest is compounded continuously with an interest rate of 5%. When will your bank account have a balance of $8,000?
In 1990, Euler Town had a population of 30,000. If the population decreases by 10% each year, during which year will the population be 10,000?
Solve Logarithmic Equations
Today’s ObjectivesSolve logarithmic equations with the
same base.Solve logarithmic equations by
rewriting in exponential form.
Answers to Last Night’s Homework
Answers to Last Night’s Homework
Type 1: Solve logarithmic equations with same base
If logbx = logby,
then x = y.
Type 3: Solve logarithmic equations with the same base
log3(5x-1) = log3(x+7) 5x – 1 = x + 7 5x = x + 8 4x = 8 x = 2
Type 3: Solve logarithmic equations with the same base
Steps:
1.If the logarithms have the same base, then drop the logarithms.
2.Solve for x
3.Check your solution.
Type 3: Solve logarithmic equations with the same base
Steps:
1.If the logarithms have the same base, then drop the logarithms.
2.Solve for x
3.Check your solution.
log4(x2-16) - log4(2x-1)=0
Steps:
1.Isolate the logarithm.
2.Rewrite in exponential form.
3.Solve for x.
4.Check for extraneous solutions.
5log5(3x + 1) + 3 = 13 5log5(3x + 1) = 10 log5(3x + 1) = 2 3x + 1 = 52
3x + 1 = 25 3x = 24 x = 8
Type 4: Solve equations with one logarithm
Steps:
1.Isolate the logarithm.
2.Rewrite in exponential form.
3.Solve for x.
4.Check for extraneous solutions.
1
2
1
2
6
6
ln 1 2 1
ln 1 2 1
ln 1 3
1ln 1 3
2ln 1 6
1
1
x
x
x
x
x
x e
x e
Type 4: Solve equations with one logarithm
Your turn! Complete these on your own
1. 7log9(x + 8) - 2 = 5
2. 9 + log3(x + 3) = 9
3. log3(9x)2 = 8
4. 5+ln(x+1)1/2=6
log5x + log(x-1)=2Type 5: Solve equations with 2+ logarithms
Steps:
1.Isolate the logarithms on one side of the equation.
2.Use properties of logarithms to condense.
3.Rewrite in exponential form.
4.Check for extraneous solutions.
ln28
2
2
ln ln(8) 2
ln ln(8) 2
ln 28
8
8
x
x
x
x
e e
xe
x e
log5x + log(x-1)=2log (5x) + log(x-1) = 2 log (5x)(x-1) = 2 log (5x2 – 5x) = 2
10log(5x -5x) = 102
5x2 - 5x = 100
x2 – x – 20 = 0 (x – 5)(x + 4) = 0 x = 5, x = –4
Type 5: Solve equations with 2+ logarithms
Steps:
1.Isolate the logarithms on one side of the equation.
2.Use properties of logarithms to condense.
3.Rewrite in exponential form.
4.Check for extraneous solutions.
Your turn! Complete these on your own
1. log 7 =2 - log x
2. log72 – log7(x-5) = 2
3. ln x + ln(7) = 3
4. 3log2x - log2(2x) = 3
EXIT CARD
Homework: Solving Logarithmic Equations Worksheet
4 4
6
9
log 3 1 log 5
log( 1) log(3) 4
10 6 55
log 9 121
x
x
x
x
Day 3: “Speed Dating Exp/Log Equations” Activity• Students will sit next to the student with the same
numbered card.• Students will become the “master” of the problem they
are given. Students can check their solution on the back of their card and work together with their partner.
• After all students have the solution to their problem, they will switch seats similar to speed dating.
• They will write down their new partner’s problem on their paper and solve it.
• If students have difficulty, they will ask their partner how to “master” the question.
• Students will switch after a given amount of time.
