1. ## Logarithimic/Exponential equations

http://img3.imageshack.us/img3/9558/img163e.jpg

1.2, 1.4, and 1.5

I need big help on 1.4 and 1.5.

I think x = 5 for 1.2. I think 1.4 and 1.5 are factorable, but I don't know how to go about doing that. No examples in my textbook...

2. Originally Posted by A Beautiful Mind
http://img3.imageshack.us/img3/9558/img163e.jpg

1.2, 1.4, and 1.5

I need big help on 1.4 and 1.5.

I think x = 5 for 1.2. I think 1.4 and 1.5 are factorable, but I don't know how to go about doing that. No examples in my textbook...
1.2 has no solution. There is no number that makes $e^{x-5} = 0$

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1.4 $e^{-x}-3e^x+2=0$

Multiply by $e^x$ : $1-3e^{2x}+2e^x = 0$

$3e^{2x}-2e^x-1= (3e^x+1)(e^x-1) = 0$

The first solution will be complex and the second real

$x = ln\left(\frac{1}{3}\right) + i\pi$ or $x = 0$

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1.5: e^{5x}-2e^{3x}-3e^x=0

Divide by e^x (see 1.2 as to why e^x=0 is not a solution)

$e^{4x}-2e^{2x}-3=0$

Set $u=e^{2x} \: \rightarrow \: u^2-2u-3=0$

$(u-3)(u+1) = 0$

$e^{2x} = 3 or e^{2x} = -1$

$2x = ln(3) \: \rightarrow \: x = \frac{1}{2}ln(3)$

OR

$2x = i\pi \: \rightarrow \: x = \frac{1}{2}i\pi$