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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 10: Q. 324 (page 1055)

In the following exercises, solve each equation.
$\frac{e^{x^{2}}}{e^{14}} = e^{5 x}$ .

Short Answer

Expert verified

The solution is $x = 7$ .

Step by step solution

01

Step 1 Given equation is,

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

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

The quotient property is given by,

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

02

Step 2 Use the quotient property in the given expression.

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

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

$x^{2} - 14 = 5 x x^{2} - 5 x - 14 = 0 (x - 7) (x + 2) = 0 x = 7, - 2$

The logarithm of negative number does not exist.

So, discard $x = - 2$ .

The solution is, $x = 7$ .

03

Step 3 Let us check the answer.

Substitute $x = 7$ into the given expression.

$\frac{e^{7^{2}}}{e^{14}} = e^{5 (7)} \frac{e^{49}}{e^{14}} = e^{35} e^{49 - 14} = e^{35} e^{35} = e^{35}$

which is true.

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