Question

In the following exercises, solve each equation.

$\frac{e^{x^2}}{e^{14}}=e^{5x}$.

Short answer

The solution is$x=7$.

Step-by-step solution

Step 1 Given equation is,

$\frac{e^{x^2}}{e^{14}}=e^{5x}$

The objective is, we need to find the solution for the given equation.

The quotient property is given by,

$\frac{a^m}{a^n}=a^{m-n}$.

Step 2 Use the quotient property in the given expression. 

$e^{x^2-14}=e^{5x}$.

The base is same. Solve equate the exponent and solve for $x$.

$$\begin{gathered} x^2-14=5x \\ {x}^2-5x-14=0 \\ \left(x-7\right)\left(x+2\right)=0 \\ x=7,-2 \end{gathered}$$

The logarithm of negative number does not exist.

So, discard $x=-2$.

The solution is,$x=7$.

Step 3 Let us check the answer.

Substitute $x=7$into the given expression.

$$\begin{gathered} \frac{e^{7^2}}{e^{14}}=e^{5\left(7\right)} \\ \frac{e^{49}}{e^{14}}=e^{35} \\ {e}^{49-14}=e^{35} \\ {e}^{35}=e^{35} \end{gathered}$$

which is true.