Equations Involving Logarithms and Exponentials

Sol A .

Sol B Dividing by 5 gives , and then taking natural logarithms of both sides gives or . Therefore , so .

Sol C Factoring gives , so either or . If , then . Since for all values of , has no solution; so is the only solution.

Sol D . (Notice that this answer checks in the original equation.)

Sol E or . Since does not satisfy the original equation, is the only answer.

Sol 1 or . Since does not check in the original equation, is the only answer.

Sol 2 or , so or .

Sol 3 or . Since does not check in the original equation, is the only answer.

Sol 4 Factoring gives , so either or . Taking natural logarithms gives or .

Sol 5 .

Sol 6 or . Since does not check in the original equation, is the only answer.

Sol 7 Dividing both sides of by 3 gives , and then taking natural logarithms of both sides gives . Then , so and therefore .

Thus , which we can also write as .