2. Chapter 5: Exponential and Logarithmic Functions
5.1 Inverse Functions
5.2 Exponential Functions
5.3 Logarithms and Their Properties
5.4 Logarithmic Functions
5.5 Exponential and Logarithmic Equations and Inequalities
5.6 Further Applications and Modeling with Exponential and Logarithmic Functions

3. 5.5 Exponential and Logarithmic Equations and Inequalities
Properties of Logarithmic and Exponential Functions
For b > 0 and b  1,
if and only if x = y.
If x > 0 and y > 0, then logb x = logb y if and only if x = y.

4. 5.5 Exponential and Logarithmic Equations and Inequalities
Type I Exponential Equations
Solved in Section 5.2
Easily written as powers of same base
i.e. 125x = 5x
Type 2 Exponential Equations
Cannot be easily written as powers of same base
i.e 7x = 12
General strategy: take the logarithm of both sides and apply the power rule to eliminate variable exponents.

5. 5.5 Type 2 Exponential Equations
Example Solve 7x = 12.
Solution

6. 5.5 Solving a Type 2 Exponential Inequality
Example Solve 7x < 12.
Solution From the previous example, 7x = 12 when
x  Using the graph below, y1 = 7x is below
the graph y2 = 12 for all x-values less than
The solution set is (–,1.277).

7. 5.5 Solving a Type 2 Exponential Equation
Example Solve
Solution
Take logarithms of both sides.
Apply the power rule.
Distribute.
Get all x-terms on one side.
Factor out x and solve.

8. 5.5 Solving a Logarithmic Equation of the Type log x = log y
Example Solve
Analytic Solution The domain must satisfy x + 6 > 0,
x + 2 > 0, and x > 0. The intersection of these is (0,).
Quotient property of logarithms
log x = log y  x = y

9. 5.5 Solving a Logarithmic Equation of the Type log x = log y
Since the domain of the original equation was (0,),
x = –3 cannot be a solution. The solution set is {2}.
Multiply by x + 2.
Solve the quadratic equation.

10. 5.5 Solving a Logarithmic Equation of the Type log x = log y
Graphing Calculator Solution
The point of intersection is at x = 2. Notice that the
graphs do not intersect at x = –3, thus supporting our
conclusion that –3 is an extraneous solution.

11. 5.5 Solving a Logarithmic Equation of the Type log x = k
Example Solve
Solution
Since it is not in the domain and must be
discarded, giving the solution set
Write in exponential form.

12. 5.5 Solving Equations Involving both Exponentials and Logarithms
Example Solve
Solution The domain is (0,).
– 4 is not valid since – 4 < 0, and x > 0.

13. 5.5 Solving Exponential and Logarithmic Equations
An exponential or logarithmic equation can be solved
by changing the equation into one of the following
forms, where a and b are real numbers, a > 0, and a  1:
a f(x) = b
Solve by taking the logarithm of each side.
loga f (x) = loga g (x)
Solve f (x) = g (x) analytically.
3. loga f (x) = b
Solve by changing to exponential form f (x) = ab.

14. 5.5 Solving a Logarithmic Formula from Biology
Example The formula gives the
number of species in a sample, where n is the number
of individuals in the sample, and a is a constant
indicating diversity. Solve for n.
Solution Isolate the logarithm and change to
exponential form.