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...
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 $\displaystyle e^{x-5} = 0$
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1.4 $\displaystyle e^{-x}-3e^x+2=0$
Multiply by $\displaystyle e^x$ : $\displaystyle 1-3e^{2x}+2e^x = 0$
$\displaystyle 3e^{2x}-2e^x-1= (3e^x+1)(e^x-1) = 0$
The first solution will be complex and the second real
$\displaystyle x = ln\left(\frac{1}{3}\right) + i\pi$ or $\displaystyle x = 0$
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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)
$\displaystyle e^{4x}-2e^{2x}-3=0$
Set $\displaystyle u=e^{2x} \: \rightarrow \: u^2-2u-3=0$
$\displaystyle (u-3)(u+1) = 0$
$\displaystyle e^{2x} = 3 or e^{2x} = -1$
$\displaystyle 2x = ln(3) \: \rightarrow \: x = \frac{1}{2}ln(3)$
OR
$\displaystyle 2x = i\pi \: \rightarrow \: x = \frac{1}{2}i\pi$