1. Solving Exponential and Logarithmic Equations
Objectives:
To solve exponential equations
To solve logarithmic equations

2. 7.6: Solve Exponential and Logarithmic Equations
Objective 1
You will be able to solve exponential equations
If the bases are the same, the moustaches are the same.

3. Exercise 1
If 2x = 23, then what is the value of x?

4. Equality of Exponentials
Property of Equality of Exponential Equations
If 𝑏 is a positive real number not equal to 1, then 𝑏𝑥=𝑏𝑦 iff 𝑥=𝑦.
2𝑥=23 iff 𝑥=3
This means that one way to solve exponential equations is to make the bases equal so that the exponents are equal.

5. Exercise 2
Solve 3 𝑥 = 𝑥+3

6. Exercise 3
Solve 9𝑥=25.

7. Taking the Log of Both Sides
Sometimes you can’t rewrite your equation so that the bases are the same (or maybe you just don’t want to). In that case, try one of the following:
Both of these are really the same thing
You may have to use the change of base formula
Take the log of both sides
Rewrite the equation in logarithmic form

8. Exercise 3 Method 1: 9 𝑥 =25 log 9 9 𝑥 = log 9 25 𝑥= log 9 25
Take the log9 of both sides. The base of the exponential becomes the base of the log.
log 𝑥 = log 9 25
𝑥= log 9 25
Use the Inverse Property to get 𝑥 on the left.
𝑥= log 25 log 9
Use the Change of Base formula.
𝑥≈1.465

9. Exercise 3 Method 1e: 9 𝑥 =25 ln 9 𝑥 = ln 25 𝑥∙ ln 9 = ln 25
Just take the natural or the common log of both sides. Pick your favorite.
ln 9 𝑥 = ln 25
𝑥∙ ln 9 = ln 25
Use the Power Property to get 𝑥 out front.
𝑥= ln 25 ln 9
Divide by ln 9 . Notice this does the Base Changing for you.
𝑥≈1.465

10. Exercise 3 Method 2: 9 𝑥 =25 log 9 25 =𝑥 𝑥= log 25 log 9 𝑥≈1.465
Write the original equation in logarithmic form
log =𝑥
𝑥= log 25 log 9
Use the Change of Base formula.
𝑥≈1.465

11. Exercise 4
Solve −3𝑒2𝑥+16=5.

12. Exercise 5 Solve each equation. 8x – 1 = 323x – 2 2x = 5 79x = 15

13. Exercise 5
Solve each equation.
8x – 1 = 323x – 2
2x = 5

14. Exercise 5
Solve each equation.
79x = 15
4e−0.3x – 7 = 13

15. Newton’s Law of Cooling
For a cooling substance (like a cup of hot chocolate) with an initial temperature of T0, the temperature T after t minutes can be modeled by T = (T0 – TR)e−rt + TR, where TR is the surrounding temperature and r is the cooling rate.

16. Exercise 6a
You warm a mug of hot chocolate to a temperature of 90°C. You leave the warm beverage on the counter while you go to tend to your wailing baby. If the hot chocolate in the mug has a cooling rate of and the temperature of the room is 20°C, how many minutes will it take for the hot chocolate to cool to a temperature of 30°C?

17. Exercise 6b
Explain how to find the cooling rate of a substance.

18. Exercise 7
A new car costs $25,000. The value of the car decreases by 15% each year. Write an exponential decay model giving the car’s value y (in dollars) after t years. In how many years with the value of the car be ½ the original value?

19. Exercise 8
According to NATO, a movie ticket to see Star Wars (Episode IV: A New Hope) in 1977 was an average price of $2.23. In 1997 when Star Wars was re-released to theaters in anticipation of the upcoming prequel trilogy, ticket prices averaged $4.59. Use the growth model y = Pert to find the annual growth rate from 1977 to 1997.

20. 7.6: Solve Exponential and Logarithmic Equations
Objective 2
You will be able to solve a logarithmic equation
“Day and Night” Escher

21. Exercise 9
If log2 x = log2 25, then what is the value of x?

22. Equality of Logarithmic Equations
Property of Equality of Logarithmic Equations
If 𝑏, 𝑥, and 𝑦 are positive real numbers with 𝑏≠1, then log𝑏 𝑥=log𝑏 𝑦 iff 𝑥=𝑦.
log2 𝑥=log2 25 iff 𝑥=25
This means that one way to solve logarithmic equations is to make the bases equal so that what you are taking the logs of are equal.

23. Exercise 10
Solve log4 (2𝑥 + 8)=log4 (6𝑥 – 12).

24. Exponentiating
Since 𝑥=𝑦 means 𝑏𝑥=𝑏𝑦, we can exponentiate both sides of an equation using the same base.
Notice the two sides of the equation become the exponents
Exponentiate means to raise a quantity to a power
Essentially rewriting a logarithmic equation as an exponential equation
Check for extraneous solutions

25. Exercise 11 Solve log7 (3x – 2) = 2. Method 1: log 7 3𝑥−2 =2
Exponentiate using 7 as the base.
3𝑥−2=49
Use the Inverse Property to simplify.
3𝑥=51
Solve for 𝑥.
𝑥=17

26. Exercise 11 Solve log7 (3x – 2) = 2. Method 2: log 7 3𝑥−2 =2 7 2 =3𝑥−2
Write in exponential form.
3𝑥−2=49
Solve for 𝑥.
3𝑥=51
𝑥=17

27. Exercise 12
Solve log6 3x + log6 (x – 4) = 2.

28. 7.6: Solve Exponential and Logarithmic Equations
Exercise 13
Solve each equation.
ln (7x – 4) = ln (2x + 11)
log2 (x – 6) = 5
log 5x + log (x – 1) = 2
log4 (x + 12) + log4 x = 3

29. Exercise 13 Solve each equation. ln (7x – 4) = ln (2x + 11)
log2 (x – 6) = 5

30. Exercise 13 Solve each equation. log 5x + log (x – 1) = 2

31. Special Cases: Quadratic
Sometimes we want to solve an equation that looks kind of like a quadratic. Well, just treat it like one.
𝑒 2𝑥 −7 𝑒 𝑥 +12=0
Let 𝑘= 𝑒 𝑥
So 𝑘 2 = 𝑒 2𝑥
𝑒 𝑥 −3 𝑒 𝑥 −4 =0
𝑒 𝑥 −3=0
𝑒 𝑥 −4=0
𝑘 2 −7 𝑘 2 +12=0
𝑒 𝑥 =3
𝑒 𝑥 =4
𝑘−3 𝑘−4 =0
𝑥= ln 3
𝑥= ln 4
𝑥≈1.099
𝑥≈1.386

32. Special Cases: Different Base
What if the exponential has different bases? Just pick one to take the log of, the other one will become an exponent. Or just take the natural log!
3 4𝑥−5 = 7 2𝑥
𝑥 4ln 3 − 2ln 7 =5 ln 3
ln 3 4𝑥−5 = ln 7 2𝑥
𝑥= 5 ln 3 4ln 3 − 2ln 7
4𝑥−5 ln 3 = 2𝑥∙ln 7
4𝑥∙ln 3 −5 ln 3 = 2𝑥∙ln 7
𝑥≈10.929
4𝑥∙ln 3 − 2𝑥∙ln 7 =5 ln 3

33. Special Cases: Different Base
What if the logarithms have different bases, one of which is a power of the other? Use a change of base—keep the smaller, get rid of the larger.
log 3 2𝑥 = log 𝑥−3
log 𝑥 2 = log 𝑥−3
log 3 2𝑥 = log 𝑥− log 3 9
4 𝑥 2 =13𝑥−3
4 𝑥 2 −13𝑥+3=0
log 3 2𝑥 = log 𝑥−3 2
4𝑥−1 𝑥−3 =0
𝑥=1/4
𝑥=3
2∙log 3 2𝑥 = log 𝑥−3

34. Solving Exponential and Logarithmic Equations
Objectives:
To solve exponential equations
To solve logarithmic equations