1. 4.6 Type 2 Exponential Equations
Solving Exponential Equations
Ones that cannot be easily written as powers of same base
i.e x = 12
General strategy: take the logarithm of both sides and apply the power rule to eliminate variable exponents
Example1 Solve 7x = 12.
Solution

2. 4.6 Solving a Type 2 Exponential Equation
Example2 Solve
Solution
Take logarithms of both sides.
Apply the power rule.
Distribute.
Get all x-terms on one side.
Factor out x and solve.

3. 4.6 Solving a Logarithmic Equation of the Type log x = log y
Example3 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

4. 4.6 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.

5. 4.6 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.

6. 4.6 Solving a Logarithmic Equation of the Type log x = k
Example4 Solve
Solution
Since it is not in the domain and must be
discarded, giving the solution set
Write in exponential form.

7. 4.6 Solving Equations Involving both Exponentials and Logarithms
Example5 Solve
Solution The domain is (0,).
– 4 is not valid since – 4 < 0, and x > 0.

8. 4.6 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.