Question

the following equation involves more than one exponential expression. solve the equation.
$6^x = 2^{x + 2}$

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 \(6^{x}=2^{x + 2}\). Using the property of logarithms \(\ln(a^{b})=b\ln(a)\), we get:
\(\ln(6^{x})=\ln(2^{x + 2})\)
\(x\ln(6)=(x + 2)\ln(2)\)

Step2: Expand the right - hand side

Expand the right - hand side of the equation:
\(x\ln(6)=x\ln(2)+2\ln(2)\)

Step3: Move terms with x to the left - hand side

Subtract \(x\ln(2)\) from both sides of the equation:
\(x\ln(6)-x\ln(2)=2\ln(2)\)

Step4: Factor out x

Factor out \(x\) from the left - hand side:
\(x(\ln(6)-\ln(2)) = 2\ln(2)\)

Step5: Solve for x

We know that \(\ln(a)-\ln(b)=\ln(\frac{a}{b})\), so \(\ln(6)-\ln(2)=\ln(\frac{6}{2})=\ln(3)\). Then we can solve for \(x\):
\(x=\frac{2\ln(2)}{\ln(6)-\ln(2)}=\frac{2\ln(2)}{\ln(3)}\)
Using a calculator, \(\ln(2)\approx0.6931\), \(\ln(3)\approx1.0986\)
\(x=\frac{2\times0.6931}{1.0986}=\frac{1.3862}{1.0986}\approx1.2619\)

Answer:

A. \(x = 1.2619\)