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Solve the exponential equation for $x: e^{x - 4} e^{2x + 9} = 14$
in Mathematics by Diamond (62,212 points) | 8 views

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Solve the exponential equation for $x: $

\[e^{x - 4} e^{2x + 9} = 14\]

Looking at the left hand side, the two terms $e^{x-4}$ and $e^{2x+9}$ both have the same base of $e$, so using the laws of exponents, we can add the powers:

\[e^{x - 4} e^{2x + 9}= e^{x-4+2x+9}=14\]

which translates into:

\[e^{3x+5}=14\]

Now because there is no common base on both sides of the equals sign, we can resort to using logarithms as an alternative method.

Apply the natural logarithm to both sides of the equation:

\[\ln{e^{3x+5}}= \ln{14}\]

\[(3x+5)\ln{e}=\ln{14}\]

We know that $\ln{e}=1$, so the equation becomes:

\[3x+5=\ln{14}\]

\[3x=\ln{14}-5\]

Therefore

\[x=\dfrac{(\ln{14}-5)}{3}\]
by Diamond (62,212 points)

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