Question

using the properties of exponents and logarithms, find the value of x in 19 + 2 ln x = 25. a. 0.05 b. 1.93 c. 3 d. 20.09

Explanation:

Step1: Isolate the logarithmic term

Subtract 19 from both sides of the equation \(19 + 2\ln x = 25\).
\(2\ln x = 25 - 19\)
\(2\ln x = 6\)

Step2: Solve for \(\ln x\)

Divide both sides by 2.
\(\ln x=\frac{6}{2}\)
\(\ln x = 3\)

Step3: Convert from logarithmic to exponential form

Recall that \(\ln x = y\) is equivalent to \(x = e^{y}\). So, for \(\ln x = 3\), we have:
\(x = e^{3}\)

Step4: Calculate the value of \(e^{3}\)

We know that \(e\approx2.71828\), so \(e^{3}\approx2.71828^{3}\approx20.0855\approx20.09\)

Answer:

D. 20.09