Question

Solve e⁽³⁻⁵ˣ⁾ = 16 for x. (You may leave e, ln, or log in your answer.)

a) x = -1/5 * ln(4/e)
b) x = 3/5 * ln(4/e)
c) x = -1/5 * ln(16/e)
d) x = 3/5 * ln(16/e)

Answer 1

The equation e^(3-5x) = 16 can be solved by taking the natural logarithm of both sides, leading to the solution x = -1/5 * ln(16/e) after simplifying and using properties of logarithms.

Step-by-step explanation:

To solve the equation e^(3-5x) = 16 for x, we start by taking the natural logarithm of both sides. The equation becomes ln(e^(3-5x)) = ln(16). Using the property that the natural logarithm and the exponential function are inverse functions, the left side simplifies to 3-5x = ln(16).

Now, solve for x by isolating it on one side which gives us x = (3 - ln(16))/5. When you evaluate ln(16), which is the power to which e must be raised to get 16, you get approximately 2.7726. Therefore, x = (3 - 2.7726)/5.

This simplifies to x = 0.04548, but since we need the answer in terms of ln or e, and according to the answer choices, we can express the equation as x = (3 - ln(16))/5. Applying properties of logarithms, we can write 16 as 42 and simplify further so that our final answer matches one of the given choices.

Thus, the answer choice that matches our calculation is x = -1/5 * ln(16/e), since ln(42/e) simplifies to 2*ln(4) - 1, which is equivalent to ln(16) - ln(e) or ln(16/e).