General solutions where f(x) = kepx (exponential types)

Principles of Exponential Equations

Exponential equations are mathematical expressions that describe growth or decay phenomena.
These equations take the form f(x) = kepx, where ‘k’ and ‘p’ are constants, ‘e’ is the base of natural logarithms (approximately equal to 2.71828), and ‘x’ is the variable.

When given an equation of the form f(x) = kepx, start solving by substituting given values (if any) into the equation.
Peel off any added or subtracted terms to the exponential term in stages until the exponential term stands alone on one side of the equation.
If you are left with an equation kepx = a, where ‘a’ is a constant, take the natural log of both sides: ln(kepx) = ln(a).
Utilising the properties of logarithms, simplify e^px to yield px = ln(a/k).
To isolate ‘x’, divide both sides of the equation by ‘p’: x = (ln(a/k))/p.

To isolate the exponential term, we divide both sides by 3, leaving e2x = 3.
Taking the natural logarithm on both sides, we get 2x = ln(3).
Lastly, we find x by dividing the equation by 2: x = (ln(3))/2.

Exponential equations are pivotal in physics, economics, biology, and many more fields.
Understanding how to solve this type of equation can help simplify problems related to exponential growth or decay, such as population growth, radioactive decay, and compound interest.