```Question 434448
{{{2(e^x +1) = 10}}}
Start by isolating the base, e, and its exponent, x. Dividing both sides by 2 we get:
{{{e^x +1 = 5}}}
Subtracting 1 from each side we get:
{{{e^x = 4}}}<br>
e and 4 are not known powers of each other. Nor are they both known powers of some third number. Because of these facts we must use logarithms to finish solving this equation. <i>Any</i> base of logarithm may be used. However, if we choose a base<ul><li>that matches the base of the exponent we will get s impler answer.</li><li>that our calculator "knows", like base 10 or base e (aka ln), we will get an answer that will be easy to convert to s decimal approximation.</li></ul>
Since base e (aka ln) logarithms fit both of the above criteria, they are clearly  the best choice. Finding the base e logarithm of each side we get:
{{{ln(e^x) = ln(4)}}}
Next we use a property of logarithms, {{{log(a, (p^q)) = q*log(a, (p))}}}, to move the exponent of the argument out in front. (It is this very property that is the reason we use logarithms on equations like these. The property lets us move the exponent, where the variable is, out in front we can now solve for the variable.) Using this property on your equation we get:
x*ln(e) = ln(4)
By definition ln(e) = 1. (This is why matching the base of the logarithm to the base of the exponent results in a simpler expression.) So thsi becomes:
x = ln(4)
This is an exact expression for the solution to your equation. If you want a decimal approximation, then use your calculator to find ln(4).```