Calculus – $a^x$ and $log_a x$

In calculus, exponential functions with a base other than e (like a^x) and logarithms with base a (like log_a x) are handled by converting them into their “natural” forms.

Since the natural exponential e^x and natural log ln(x) have very simple derivatives, we use them as the bridge to understand a^x and log_a x.

1. The General Exponential Function: a^x

To work with a^x in calculus, we rewrite it using the identity a = e^{ln a}. This gives us:

The Derivative

Using the Chain Rule on e^{x ln a}, where ln(a) is just a constant:

2. The General Logarithmic Function: log_a x

For logarithms with base a, we use the Change of Base Formula to convert the expression into natural logarithms:

The Derivative

Since 1 / (ln a) is a constant, we only need to differentiate ln(x):