Question

solve the equation. give an exact solution, and also an approximate solution to four decimal places. 5^{3x - 2}=17 a. the exact solution is x = . (use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Take the logarithm of both sides

Take the natural - logarithm of both sides of the equation \(5^{3x - 2}=17\). Using the property \(\ln a^{b}=b\ln a\), we get \((3x - 2)\ln5=\ln17\).

Step2: Solve for \(x\)

First, expand the left - hand side: \(3x\ln5-2\ln5=\ln17\). Then, isolate the term with \(x\): \(3x\ln5=\ln17 + 2\ln5\). Since \(2\ln5=\ln5^{2}=\ln25\), we have \(3x\ln5=\ln17+\ln25\). Using the property \(\ln a+\ln b=\ln(ab)\), we get \(3x\ln5=\ln(17\times25)=\ln425\). So, \(x=\frac{\ln425}{3\ln5}\).

Step3: Calculate the approximate value

We know that \(\ln425\approx6.0544\), \(\ln5\approx1.6094\). Then \(3\ln5\approx4.8282\). So, \(x=\frac{\ln425}{3\ln5}\approx\frac{6.0544}{4.8282}\approx1.2539\).

Answer:

The exact solution is \(x = \frac{\ln425}{3\ln5}\), and the approximate solution to four decimal places is \(x\approx1.2539\).