Question

find an approximate solution to the equation.
(0.87)^x = 5

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

a. the solution is x ≈
(do not round until the final answer. then round to four decimal
b. the solution is not a real number.

Explanation:

Step1: Take logarithms on both sides

To solve the equation \((0.87)^x = 5\), we can take the natural logarithm (or common logarithm) of both sides. Using the natural logarithm, we get \(\ln((0.87)^x)=\ln(5)\).

Step2: Apply the logarithm power rule

By the power rule of logarithms, \(\ln(a^b)=b\ln(a)\), so the left - hand side becomes \(x\ln(0.87)=\ln(5)\).

Step3: Solve for \(x\)

We can solve for \(x\) by dividing both sides of the equation \(x\ln(0.87)=\ln(5)\) by \(\ln(0.87)\). So \(x = \frac{\ln(5)}{\ln(0.87)}\).

Step4: Calculate the value

First, we know that \(\ln(5)\approx1.6094\) and \(\ln(0.87)\approx - 0.1394\). Then \(x=\frac{1.6094}{- 0.1394}\approx - 11.5452\).

Answer:

A. The solution is \(x\approx - 11.5452\)