Question

the following equation involves more than one exponential expression. solve the equation.
$2^{7x + 2} = 5^x$

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. $x = \square$
(do not round until the final answer. then round to four decimal places as needed.)

b. there is no solution.

Explanation:

Step1: Take natural log on both sides

Take the natural logarithm of both sides of the equation \(2^{7x + 2}=5^{x}\). Using the property \(\ln(a^{b})=b\ln(a)\), we get:
\(\ln(2^{7x + 2})=\ln(5^{x})\)
\((7x + 2)\ln(2)=x\ln(5)\)

Step2: Expand and rearrange terms

Expand the left - hand side: \(7x\ln(2)+2\ln(2)=x\ln(5)\)
Move all terms with \(x\) to one side: \(7x\ln(2)-x\ln(5)=- 2\ln(2)\)
Factor out \(x\) from the left - hand side: \(x(7\ln(2)-\ln(5))=-2\ln(2)\)

Step3: Solve for x

We can solve for \(x\) by dividing both sides by \((7\ln(2)-\ln(5))\):
\(x=\frac{- 2\ln(2)}{7\ln(2)-\ln(5)}\)
First, calculate the values of the logarithms:
\(\ln(2)\approx0.6931\), \(\ln(5)\approx1.6094\)
\(7\ln(2)-\ln(5)=7\times0.6931 - 1.6094=4.8517-1.6094 = 3.2423\)
\(-2\ln(2)=-2\times0.6931=-1.3862\)
Then \(x=\frac{-1.3862}{3.2423}\approx - 0.4275\)

Answer:

A. \(x = - 0.4275\)