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# 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.

![](https://3104074691-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LmGGOR79oy-_JywJoK5%2F-M29jhpp5zH6OtYxa_4g%2F-M29jiKi79H2t-UE3Rn4%2Fexponential-and-inverse.png?generation=1583949474632716\&alt=media)

## 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.