1. Chapter 6 Exponential and Logarithmic Functions and Applications Section 6.4

2. Section 6.4 Exponential and Logarithmic Equations
Solving Exponential Equations
Exponential Equality Property
Base 10 or Base e
Graphically
Fundamental Properties of Logarithms
Solving Exponential Equations Using Properties of Logarithms
Solving Logarithmic Equations
Solving Literal Equations Involving Exponential and Logarithmic Equations
Change of Base Formula
Applications

3. For b > 0 and b  1, if bm = bn, then m = n.
Exponential Equality Property:
For b > 0 and b  1, if bm = bn, then m = n.
*Solving Exponential Equations with the Exponential Equality Property:
Express each side of the equation as a power
of the same base.
Apply the exponential equality property to equate
the exponents.
3. Solve for the variable.
4. Check your solution.
*Also referred to as solving by "equating the bases" or "relating the bases."

4. Use the exponential equality property to solve 292x + 4 = 162.
Step 1. Express each side of the equation as a power of the
same base.
First, we divide both sides of 292x + 4 = 162 by “2.”
92x + 4 = 81
Next, we express each side of the equation as a power of “3.”
(32)2x + 4 = 34
34x + 8 = 34
Make sure to distribute correctly: 2(2x + 4) = 4x + 8
Steps 2-3. Apply the exponential equality property to equate the
exponents. Solve for the variable.
4x + 8 = 4
x = –1
Step 4. Check your solution.
29[2(–1) + 4] =  292 =  2(81) = True

5. Use the exponential equality property to solve
Check:

6. 2. Convert the equation to logarithmic form.
Solving Exponential Equations with Base 10 or Base e.
Isolate the power (the term containing the variable exponent) on one side of the equation.
If necessary, divide both sides of the equation by any coefficient of the power term.
2. Convert the equation to logarithmic form.
3. Solve for the variable. Use a calculator if necessary.
4. Check your solution.

7. Solve the exponential equation 2(–6 + 102x) = 16. 26
Solve the exponential equation 2(– x) = Round your answer to 4 decimal places as needed.
Step 1. Isolate the power.
First, divide both sides of the equation by “2.” Then, add “6” to
both sides of the equation.
– x = 8.13
102x = 14.13
Step 2. Convert the equation to logarithmic form.
log = 2x or log = 2x
Step 3. Solve for the variable.
x = log 14.13 2
x 
Step 4. Check your solution.
2[–6 + 10(2 )] =  2(8.1319) =  16.26

8. Solve the exponential equation 25. 6 = –2 + 3. 1e–0. 15x
Solve the exponential equation 25.6 = – e–0.15x. Round your answer to 4 decimal places as needed.
27.6 = 3.1e–0.15x
= e–0.15x
loge = –0.15x or ln = –0.15x
x = ln
–0.15
x  –
You can check the answer.

9. Can use intersection method or x-intercept method.
Solving Exponential Equations Graphically
Use your graphing calculator to solve 2x + 5 = 32x + 6 – 1 graphically.
Can use intersection method or x-intercept method.
Let Y1 = 2x + 5 and Y2 = 32x + 6 – 1. We will use the window
[–6, 6, 1] by [– 6, 12, 1] and find the intersection.
The answer to the given equation is x = –2.
Checking: 2(–2 + 5) = 3[2(–2) + 6] – 1
8 = 8

10. Note: These properties apply to natural logarithms as well.
Recall the Basic Properties of Logarithms: For b > 0 and b  1,
logb 1 = 0, logb b = 1, logb bx = x, and
Fundamental Properties of Logarithms
For positive real numbers M, N, and b, b  1, and any real number k:
Product Property:
Quotient Property:
Power Property:
Note: These properties apply to natural logarithms as well.

11. = log3 (47) = log3(4) + log3(7) = 1.2619 + 1.7712 = 3.0331
Use the fact that log3 4 = and log3 7 = and the properties of logarithms to estimate the value of the following expressions. Round your answers to 4 decimal places as needed.
a. log3 28
= log3 (47) = log3(4) + log3(7) = =
b. log3 (1.75)
= log3 (7/4) = log3(7) – log3(4) = – = c.

12. a. 5 log x + log 9 = log x5 + log 9 = log (9x5)
Use the properties of logarithms to rewrite each expression as a single logarithm.
a. 5 log x + log 9
= log x5 + log 9 = log (9x5)
b. log (x + 3) – log (x2 – 9) c. or

13. Expand the given expression in terms of simpler logarithms
Expand the given expression in terms of simpler logarithms. Assume that all variable expressions are positive real numbers.

14. Solving Exponential Equations Using Properties of Logarithms
1. Isolate the power on one side of the equation.
2. Take the logarithm of both sides of the equation; may
take either the common logarithm (base 10) or a
natural logarithm (base e).
3. Apply the power property of logarithms to simplify
(that is, "bring down" the variable exponent to the
front).
4. Solve for the variable.
5. Check your solution.

15. Step 1: Isolate the power on one side of the equation. 4x = 12
Solve the exponential equation –7 + 4x = 5. Round your answer to 4 decimal places.
Step 1: Isolate the power on one side of the equation.
4x = 12
Step 2: Take the logarithm of both sides of the equation.
log 4x = log 12
Step 3: Apply the power property of logarithms.
x log 4 = log 12
Step 4: Solve for the variable.
5. Check your solution.
–7 + 4(1.7925)  5

16. Solve the exponential equation 27ex – 2 = 7
Solve the exponential equation 27ex – 2 = 7. Round your answer to 4 decimal places.
ex – 2 =
ln ex – 2 = ln
(x – 2) ln e = ln
(x – 2) (1) = ln
(Recall ln e is equivalent to loge e)
x = ln
x 
Checking:
27e( – 2)  7

17. For positive real numbers m, n, and b, b  1,
Logarithmic Equality Property:
For positive real numbers m, n, and b, b  1,
if logb m = logb n, then m = n.
Solving Logarithmic Equations:
1. Isolate the logarithmic expression on one side of
the equation. If needed, apply the properties of
logarithms to combine all logarithms as a single
logarithm.
2. Convert the logarithmic equation to an exponential
equation.
3. Solve for the variable.
4. Check for possible extraneous solutions.

18. Solve the logarithmic equation 6 log 4x = 18.
Step1. Isolate the logarithmic expression on one side of the
equation.
log 4x = 3
Step 2: Convert the logarithmic equation to an exponential
103 = 4x
Step 3: Solve for the variable.
4x = 1000
x = 250
Step 4: Check for possible extraneous solutions.
6 log [4(250)] = 18
= 18

19. Solve the logarithmic equation
We discard x = –10/3 (it is not in the domain). Thus, x = 10.
You can verify the solution.

20. Solve the equation ln x + ln (x + 3) = ln (x + 15).
ln [x(x + 3)] = ln (x + 15)
ln (x2 + 3x) = ln (x + 15)
x2 + 3x = x  Logarithmic equality
x2 + 3x – x – 15 = 0
x2 + 2x – 15 = 0
(x + 5)(x – 3) = 0
x = –5 or x = 3
We discard x = –5 (it is not in the domain). Thus, x = 3.
You can verify the solution.

21. Recall ln (2x – 1) = 3 is equivalent to loge(2x – 1) = 3 e3 = 2x – 1
Solve the equation ln (2x – 1) = 3 and approximate your
answer to 4 decimal places.
Recall ln (2x – 1) = 3 is equivalent to loge(2x – 1) = 3
e3 = 2x – 1
2x = e3 + 1
2x =
x 

22. Solve for y: Solving Literal Equations Involving Exponential or
Logarithmic Equations
Solve for y:

23. Change of Base Formula
For positive real numbers M, a, and b, a  1, b  1,
We may convert any given base into either base 10 or base e.
Example: Use the change of base formula to evaluate log2 5, and graph y = log2 x by applying the change of base formula. or
Graph of y = log2 x:

24. The function P(t) = (3.2)t models the number of digital 3D screens worldwide for t number of years after Using this model, estimate when the number of digital 3D screens worldwide reached approximately 9,000. Solve algebraically and round your answer to the nearest whole number. Source: mpaa.org.
The number of digital 3D screens reached approximately 9,000 in
2009.

25. The number of prohibited firearms intercepted by the Transportation Security Administration (TSA) at U.S. airport screenings from 2005 to 2009 can be modeled by the function
f(x) = – ln x, where x = 1 represents 2005,
x = 2 is 2006, and so on. Use this model to estimate when the
TSA intercepted approximately 892 firearms. Round your answer to the nearest whole number. Sources:
The TSA intercepted approximately 892 firearms in 2009.

26. Using your textbook, practice the problems assigned by your instructor to review the concepts from Section 6.4.