Question

(a) options: a x=-1, b x=0, c x=1, d x=2 (b) ln(x) = 7 options: a x=2.718, b x=19.0, c x = ln(e) (which is 1), d x = e^7 (which is about 1096.6)

Explanation:

Step1: Recall the definition of natural logarithm

The natural logarithm function \(\ln(x)\) is the inverse of the exponential function \(e^x\). By the definition of inverse functions, if \(\ln(x) = y\), then \(x = e^y\).

Step2: Apply the definition to the equation \(\ln(x)=7\)

Using the relationship between the natural logarithm and the exponential function, when \(\ln(x) = 7\), we can rewrite this equation in exponential form. Substituting \(y = 7\) into \(x = e^y\), we get \(x=e^{7}\). Calculating \(e^{7}\) (where \(e\approx2.718\)), we have \(e^{7}\approx1096.6\) (or approximately \(1096\) depending on the level of approximation). Now let's check the options:

• Option A: \(x = 2.718\) is approximately \(e\), not \(e^{7}\), so this is incorrect.
• Option B: \(x = 19.0\) is not equal to \(e^{7}\), so this is incorrect.
• Option C: \(x=\ln(e)=1\) is not related to \(\ln(x) = 7\), so this is incorrect.
• Option D: \(x = e^{7}\) (which is about \(1096.6\)) matches our result from the exponential form conversion, so this is correct.

Answer:

D. \(x = e^{7}\) (which is about \(1096.6\))