Find \(x\) if \(2e^x = 10\).

To solve the equation (2e^x = 10), the first step is to isolate the exponential term. We can start by dividing both sides of the equation by 2:

[

e^x = \frac{10}{2} = 5.

]

Now that we have (e^x = 5), we can eliminate the exponential by taking the natural logarithm of both sides. The natural logarithm, (\ln), is the inverse function of the exponential function (e^x). Thus, we apply the logarithm:

[

\ln(e^x) = \ln(5).

]

By the properties of logarithms, specifically that (\ln(e^x) = x), we can simplify the left side:

[

x = \ln(5).

]

Therefore, the value of (x) that satisfies the original equation (2e^x = 10) is (\ln(5)). This solution correctly finds (x) while ensuring that all necessary steps highlight the relationship between exponential and logarithmic functions.