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Numerade Educator

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Problem 54 Hard Difficulty

Solve each equation for x.

(a) $ \ln (\ln x) = 1 $
(b) $ e^{ax} = Ce^{bx} $ , where $ a \neq b $

Answer

a) $x=e^{e}$
b) $x=\frac{\ln C}{a-b}$

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LM

Lyka M.

September 26, 2021

adding functions

Video Transcript

all right. Since we have a lager with Mickey equation here, what we're going to want to do is change it to exponential form in order to solve it. So remember, when we see natural log, it means log Basie. So we have logged Basie of the natural log of X equals one. So changing that to exponential form will look like this. He to the first power equals natural log of X. And each of the first power is just eat. So now we hav e equals natural log of X. Well, let's do it again. Let's think of the natural August Log Base E sweetie equals log Basie of X. And let's change that into its exponential form. We have e to the power e equals X exes e to the power. And for part B, we want to solve for X notice that we have X in two different places. So we're going to want to combine those in some way. So let's divide both sides of the equation by e to the B X. So we have each of the A f divided by e to the B X equal C. Okay. Remember your exponents rules If you have based to a power divided by the same base to a power, you could subtract those powers so we would have e to the power of a X minus. B x equal C. Okay, now we can take the natural log of both sides. So when we take the natural log of the left, we get a X minus B X, and when we take the natural log of the right, we get natural log of seat. So let's factor a out of the left side. Excuse me. Last factor X out of the left side and divide both sides by a minus bi, and we will have x by itself.