Question

solve for x.
ln4 + ln(x - 3) = -1
do not round any intermediate computations, and round your answer to the nearest hundredth.
x =

Explanation:

Step1: Use logarithm property

Using the property of logarithms \(\ln a+\ln b = \ln(ab)\), we can rewrite the left - hand side of the equation \(\ln4+\ln(x - 3)\) as \(\ln[4(x - 3)]\). So the equation becomes \(\ln[4(x - 3)]=- 1\).

Step2: Convert logarithmic to exponential form

Recall that if \(\ln y = k\), then \(y = e^{k}\) (since the natural logarithm \(\ln\) has a base of \(e\)). Applying this to our equation \(\ln[4(x - 3)]=-1\), we get \(4(x - 3)=e^{-1}\).

Step3: Solve for \(x\)

First, divide both sides of the equation \(4(x - 3)=e^{-1}\) by 4: \(x - 3=\frac{e^{-1}}{4}\).
Then, add 3 to both sides of the equation: \(x = 3+\frac{1}{4e}\).
We know that \(e\approx2.71828\), so \(e^{-1}=\frac{1}{e}\approx\frac{1}{2.71828}\approx0.367879\).
Then \(\frac{e^{-1}}{4}\approx\frac{0.367879}{4}\approx0.09196975\).
And \(x=3 + 0.09196975\approx3.09\) (rounded to the nearest hundredth).

Answer:

\(x\approx3.09\)