Question

given the functions ( f(x) ) and ( g(x) ) below, find all solutions to the equation ( f(x)=g(x) ) graphically to the nearest hundredth.

( f(x)=8 ln x - 4 )

( g(x)=8 )

Explanation:

Step1: Set up the equation

Set \(8\ln x - 4=8\).

Step2: Isolate the logarithmic term

Add \(4\) to both sides: \(8\ln x=8 + 4=12\). Then divide by \(8\): \(\ln x=\frac{12}{8}=\frac{3}{2}\).

Step3: Solve for \(x\) using the property of logarithms

Since \(\ln x = a\) implies \(x = e^{a}\), here \(x = e^{\frac{3}{2}}\). Calculate \(e^{\frac{3}{2}}\approx4.48\) (using a calculator, \(e\approx2.71828\), \(e^{1.5}=e^{\frac{3}{2}}\approx4.48\)).

Answer:

\(x\approx4.48\)