How can logarithms be used to solve exponential equations?

Question

Elijah Allen · Accepted Answer

The obvious use is for equations of different bases, such as for example #2^(f(x)) = 3^(g(x))# You can never get 3 and 2 on the same base without the use of logarithms, so to solve this equation for #x#, you need to use logarithms, so it becomes* #f(x)*ln(2) = g(x)*ln(3)# And from then on, just solve the equation using algebra, however, what most people forget is that even when you have the same base you're still using logarithms, it's just a step we often skip. For example: #2^(f(x)) = 2^(g(x))#
#f(x)*ln(2) = g(x)*ln(2)#
#f(x) = g(x)# We're used to going from the first one to the third skipping the second so we forget it's there. Logs also have more uses than just exponential equations but those were mostly important back in the days before the calculator, in which it made calculating stuff much, much, simpler. *I've used the natural log because it's convenient in Calculus and it has less characters to type here, but any base could be used.

Madelyn Anderson · Answer

undefined Logarithms can be used to solve exponential equations by converting them into simpler forms. When you have an equation in the form $a^x = b$, where $a$ and $b$ are constants and $x$ is the variable, you can use logarithms to solve for $x$. Taking the logarithm of both sides of the equation allows you to isolate $x$. Depending on the base of the exponential equation, you may use different types of logarithms (e.g., common logarithms, base-10 logarithms, natural logarithms with base $e$). Once you apply the logarithm, you can solve for $x$ using algebraic manipulation.