The simple formula that describes exponential growth is

$y = a^{x}$

Where $a$ is a constant that depends on the system being studied, and $x$ is the exponent also called the index.

Taking the logarithm of both sides of the general equation gives

So, if we plot $\log y$ against $x$, an exponential relationship will plot as a straight line with a gradient of $\log a$.

The more general formula for exponential growth can be written in the form

$y = ae^{bx}$

where $a$ and $b$ are constants that depends on the system.

Taking the natural logarithm of both sides of the this equation gives:

using logarithm rules for multiplication, this is the same as

however we know that $\ln e =1$,

so $ln y = bx + ln a$