Final answer:
To make x the subject of the equation y = ae^x, we apply the natural logarithm to both sides, use logarithmic properties to separate terms, and finally isolate x.
Step-by-step explanation:
The student has asked to make x the subject of the equation y = ae^x, where e is the base of the natural logarithm, often written as ln (the inverse of e^x). To isolate x, we can apply the natural logarithm to both sides of the equation, which allows us to use the property that the natural log and the exponential function are inverse functions of each other.
Step-by-Step Solution:
Take the natural logarithm of both sides: ln(y) = ln(ae^x).
Apply the logarithmic property: ln(a) + ln(e^x) (because ln(xy) = ln(x) + ln(y)).
Since ln(e^x) = x and ln(a) is a constant, we have ln(y) = ln(a) + x.
Subtract ln(a) from both sides to solve for x: x = ln(y) - ln(a).