Exponential Computations

note: migrated 18 Mar 2021

Exponential Computations

Forward

Often an exponential is used to represent a population or an amount of something.

If we want to find the amount or population at a certain time or other independent variable, we substitute our variables and compute the amount.

$$population = P_0 e^{at}$$

• $P_0$ is the population when $t=0$.

• $a$ tells us how fast the population is growing.

• $t$ is the time.

• We have to be certain that the units for $a$ and $t$ match.

Inverse or Logarithm

If we want to find the time at which a population reaches a certain number (dependent variable), we must use the inverse of the exponential. This inverse is the logarithm.

Brute force inverse

To find the inverse, you can also guess and adjust your number for the independent variable until you match the dependent variable.

Graphical Explanation

To go up and to the left, we use the exponent.

To go right and down, we are using the inverse.

Inverses

Note that the logarithm and exponential are inverses of each other.

$\log e^x = x$

$e^{\log x} = x$

Where in this case, log means the natural log.

Inverting a Logarithm

You may have the equation below and want to find t.

$y=A e^{b\cdot t}$

Your strategy is to manipulate the equation so that you can use the inverse.

$\frac{y}{A} = e^{b \cdot t}$

At this point we can take the logarithm of both sides since the logarithm is the inverse of the exponential function.

$\log(y/A) = log(e^{b \cdot t}) = b \cdot t$

$t = \frac{\log(y/A)}{b}$

Computation

Note that the natural log is used to invert $e^x$, while the base 10 log is used to invert $10^x$.

On calculators and computers, the natural log is referred to as ln or log while the base 10 log is referred to as log or log10.

Be sure you have tested your functions and know which one to use.