Below is a plot of the curve $y=2^x$ with the tangent to the curve drawn.

If you look closely, you will see that the gradient of the tangent is smaller than the $y$ coordinate of the point. That is, the gradient of the tangent is smaller than $2^x.$

Here is $y=3^x,$ and if you look carefully you will see that the gradient is always larger than $3^x$:

This suggests there should be a number between $2$ and $3$ where the gradient is always equal to the value of the function.

This number is called $e,$ and

Here are some facts about $e$:

• $e$ is an irrational number

• The gradient of $y=e^x$ is equal to $e^x$, as you can see below:

• When we take a logarithm with base $e,$ instead of ${\log }_e(x)$ we usually write

  $$\ln (x)$$

  The $n$ stands for "natural," and we call this the natural logarithm

• $e$ was first discovered in the context of lending money with interest, as you can read about here

1. Solve the equation

   $$2^x{e}^{2x}=e^5$$

2. Solve the equation

   $$\ln (x+1)-\ln (x-1)=2$$

3. Solve the equation

   $$\ln (x+2)+\ln (2x+1)=\ln (5)$$