Find a solution for the equation e^2x + 3*e^x = 10

Monday, July 27, 2015

The equation $\displaystyle{{e}}^{{{2}{x}}}+{3}\cdot{{e}}^{{x}}={10}$ has to be solved for
x.

$\displaystyle{{e}}^{{{2}{x}}}$ can be rewritten as $\displaystyle{{\left({{e}}^{{x}}\right)}}^{{2}}$
.

The equation now be written
as:

$\displaystyle{{\left({{e}}^{{x}}\right)}}^{{2}}+{3}\cdot{{e}}^{{x}}={10}$

$\displaystyle{{\left({{e}}^{{x}}\right)}}^{{2}}+$
3*e^x - 10 = 0

$\displaystyle{{\left({{e}}^{{x}}\right)}}^{{2}}+{5}\cdot{{e}}^{{x}}-{2}{{e}}^{{x}}-{10}=$
0

$\displaystyle{\left({{e}}^{{x}}\right)}{\left({{e}}^{{x}}+{5}\right)}-{2}{\left({{e}}^{{x}}+{5}\right)}=$
0

$\displaystyle{\left({{e}}^{{x}}-{2}\right)}{\left({{e}}^{{x}}+{5}\right)}={0}$

$\displaystyle{{e}}^{{x}}-{2}=$
0

$\displaystyle{{e}}^{{x}}={2}$

x = ln
2

$\displaystyle{{e}}^{{x}}+{5}={0}$

$\displaystyle{{e}}^{{x}}=$
-5

This is not possible as e is a positive number and the power of
a positive number is always positive.

The solution of the equation
$\displaystyle{{e}}^{{{2}{x}}}+{3}\cdot{{e}}^{{x}}={10}$ is x = ln 2

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