4.6 Type 2 Exponential Equations Solving Exponential Equations Ones that 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 Example1 Solve 7x = 12. Solution

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.

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

4.6 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.

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.

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.

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.

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.