Question

How can I find the x here: (e^-x)*x=0.3085?

a) Solve using the natural logarithm.
b) Apply the quadratic formula.
c) Use the power rule for differentiation.
d) Substitute various values of x to approximate.

Answer 1

To solve (e^-x)*x = 0.3085 using the natural logarithm, isolate e^-x, take the natural logarithm of both sides, simplify, and then solve numerically for x because it also appears in the logarithm's argument.

Step-by-step explanation:

The equation given is (e^-x)*x = 0.3085. To solve for x, we should use the natural logarithm because it is the inverse function of the exponential function.

Here are the steps you can follow:

1. First, isolate the exponential term by dividing both sides of the equation by x, assuming x is not equal to 0. This gives us e^-x = 0.3085 / x.
2. Next, take the natural logarithm of both sides to get -x = ln(0.3085 / x).
3. After simplifying, we have x = -ln(0.3085 / x).
4. This equation now needs to be solved numerically since the x is also in the denominator inside the logarithm.
5. You can use a calculator to substitute various values for x until you find a value that satisfies the equation, or you can use numerical methods such as the Newton-Raphson method to find a more precise solution.

Applying the quadratic formula, power rule for differentiation, or substituting various values for x without taking the logarithm will not work in this case because the equation involves both an exponential and a linear term.