1. Exponential Equations
An exponential equation is one in which the variable occurs in the exponent. For example,
2x = 7
The variable x presents a difficulty because it is in the exponent.
To deal with this difficulty, we take the logarithm of each side and then use the Laws of Logarithms to “bring down x” from the exponent.

2. Exponential Equations
ln 2x = ln 7
x ln 2 = ln 7
 2.807
Given Equation
Take In of each side
Law 3 (Bring down exponent)
Solve for x
Calculator

3. Exponential Equations
Recall that Law 3 of the Laws of Logarithms says that
loga AC = C loga A.
The method that we used to solve 2x = 7 is typical of how we solve exponential equations in general.

4. Example 1 – Solving an Exponential Equation
Find the solution of the equation 3x + 2 = 7, rounded to six decimal places.
Solution:
We take the common logarithm of each side and use Law 3.
3x + 2 = 7
log(3x + 2) = log 7
Given Equation
Take log of each side

5. Example 1 – Solution (x + 2)log 3 = log 7 x + 2 =  –0.228756 cont’d
Law 3 (bring down exponent)
Divide by log 3
Subtract 2
Calculator

6. Example 1 – Solution Check Your Answer
cont’d
Check Your Answer
Substituting x = – into the original equation and using a calculator, we get
3(– ) + 2  7

7. Example 4 – An Exponential Equation of Quadratic Type
Solve the equation e2x – ex – 6 = 0.
Solution:
To isolate the exponential term, we factor.
e2x – ex – 6 = 0
(ex)2 – ex – 6 = 0
Given Equation
Law of Exponents

8. Example 4 – Solution (ex – 3)(ex + 2) = 0 ex – 3 = 0 or ex + 2 = 0
cont’d
(ex – 3)(ex + 2) = 0
ex – 3 = or ex + 2 = 0
ex = ex = –2
The equation ex = 3 leads to x = ln 3.
But the equation ex = –2 has no solution because ex > 0 for all x.
Thus, x = ln 3  is the only solution.
Factor (a quadratic in ex)
Zero-Product Property

9. Logarithmic Equations

10. Logarithmic Equations
A logarithmic equation is one in which a logarithm of the variable occurs.
For example,
log2(x + 2) = 5
To solve for x, we write the equation in exponential form.
x + 2 = 25
x = 32 – 2 = 30
Exponential form
Solve for x

11. Logarithmic Equations
Another way of looking at the first step is to raise the base, 2, to each side of the equation.
2log2(x + 2) = 25
x + 2 = 25
x = 32 – 2 = 30
Raise 2 to each side
Property of logarithms
Solve for x

12. Logarithmic Equations
The method used to solve this simple problem is typical. We summarize the steps as follows.

13. Example 6 – Solving Logarithmic Equations
Solve each equation for x.
(a) ln x = 8
(b) log2(25 – x) = 3
Solution:
(a) ln x = 8
x = e8
Therefore, x = e8  2981.
Given equation
Exponential form

14. Example 6 – Solution We can also solve this problem another way:
cont’d
We can also solve this problem another way:
ln x = 8
eln x = e8
x = e8
Given equation
Raise e to each side
Property of ln

15. Example 6 – Solution
cont’d
(b) The first step is to rewrite the equation in exponential form.
log2(25 – x) = 3
25 – x = 23
25 – x = 8
x = 25 – 8 = 17
Given equation
Exponential form (or raise 2 to each side)

16. Example 6 – Solution Check Your Answer If x = 17, we get
cont’d
Check Your Answer
If x = 17, we get
log2(25 – 17) = log2 8 = 3