Equations Involving Logarithms and Exponentials

Recall that if , then

.

(Notice that is undefined if .)

We will denote natural logarithms (logarithms with base ) by .

Furthermore, .

Ex 1 Solve the equation .

Sol or . Since does not check in the original equation, is the only solution.

Ex 2 Solve the equation , and write your answer using natural logarithms.

Sol Dividing by 6 gives , and then taking natural logarithms of both sides gives or . Then , so and therefore . Since , we can also write the answer as .

Pr A Solve the equation .

Pr B Solve the equation , writing your answer using natural logarithms.

Pr C Find all solutions of .

Pr D Solve the equation .

Pr E Find all solutions of .

Pr 1 Solve the equation .

Pr 2 Solve the equation .

Pr 3 Solve the equation .

Pr 4 Solve the equation .

Pr 5 Solve the equation .

Pr 6 Solve the equation .

Pr 7 Solve the equation , and write your answer using natural logarithms.