Question

the following equation involves a single exponential expression. solve the equation.
$(9.7)^{2x} = 11$

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 log of both sides

Take the natural logarithm (ln) of both sides of the equation \((9.7)^{2x} = 11\). By the logarithm power rule \(\ln(a^b)=b\ln(a)\), we get \(2x\ln(9.7)=\ln(11)\).

Step2: Solve for x

First, isolate \(x\) by dividing both sides by \(2\ln(9.7)\). So \(x = \frac{\ln(11)}{2\ln(9.7)}\). Now calculate the values: \(\ln(11)\approx2.397895\), \(\ln(9.7)\approx2.272128\). Then \(2\ln(9.7)\approx4.544256\). Divide \(\ln(11)\) by \(2\ln(9.7)\): \(x=\frac{2.397895}{4.544256}\approx0.5277\).

Answer:

A. \(x = 0.5277\)