Solving Exponential Equations

Since we now know about exponential equations and their inverses, the natural log, we can easily solve for any variable in an exponential function. It's all about plugging in values for what we do know and solving for what we don't know. If the variable we are solving for happens to be in an exponent, we can apply the natural log to each side of the equation to help us solve.

Here is the general form of solving exponential equations:

y = e^x

Now to get x out of the exponential function, we will apply the natural log to each side.

y = e^x
ln(y) = ln(e^x)
ln y = x
x = ln y

Sample Problem

Let's go back to our bacteria problem.

Just for fun, let's revisit the water fountain bacteria problem we talked about in the Section "The e^x Function." A bacteria is growing on a water fountain continuously and the number of bacteria is defined by the equation f(t) = 200e^0.415t where t is the time in hours, and f(t) is the number of bacteria. How many hours will it take for the bacteria to triple?

Since we started with 200 bacteria, it would triple to 600 bacteria. We will plug those numbers into our equation to see what is left to solve for. (Glade commercial playing in the background: "Plug it in, Plug it in.")

f(t) = 200e^0.415t
600 = 200e^0.415t

At this point we need to divide each side by 200. Our goal is to get t by itself. It's not very sociable.

Now this is where our previous knowledge of inverses come into play. Take the natural log of both sides to get rid of the e function which seems to be holding our variable t hostage.

Sample Problem

The world population grows continuously at about 1.915% a year. Due to the normal death rates, the overall population growth stays around 0.5%. Barring any population threat, like a zombie virus or some other pandemic, find how long it would take for the world population to double. Our equation will be P = e^0.005t.

Since, we don't know exactly how many people are in the world, we can always make up some numbers, because in the end it's the final population divided by the initial population anyways. Let's assume there are 7 billion people in the world.

We may need some zombies if we have that many people living on earth. Experts say that they feel like population will stop growing and start declining, so we may be safe from over population.

Sample Problem

What if we had a natural log function? We apply the same rules.

Solve ln(x²) = 121

This time we can raise each side of our equation as an exponent with base e since e is the inverse of ln.