Question

solve. write your answer in simplest form using integers, fractions, and natural logarithms. 65 = e^x - 2 x =

Explanation:

Step1: Isolate the exponential term

We start with the equation \(65 = e^{x}-2\). To isolate \(e^{x}\), we add 2 to both sides of the equation.

$$ 65 + 2=e^{x}-2 + 2 $$

Simplifying the left and right sides, we get \(67 = e^{x}\).

Step2: Apply the natural logarithm

To solve for \(x\) when we have \(e^{x}=67\), we take the natural logarithm (ln) of both sides. Recall that \(\ln(e^{a})=a\) by the property of logarithms (the natural logarithm and the exponential function with base \(e\) are inverse functions). So we have:

$$ \ln(e^{x})=\ln(67) $$

Using the property \(\ln(e^{x}) = x\), we find that \(x=\ln(67)\).

Answer:

\(x = \ln(67)\)