Log/Exponential Functions

Logarithmic and Exponential Functions

An exponential function is a function in which the variable appears in the exponent. A common example is:

$$f(x) = 2^x$$

where the base ($2$) is raised to the power of $x$.

The natural logarithm function, denoted as $\ln(x)$, is the inverse of the exponential function with base e.

The key differentiation rules for logarithmic and exponential functions are as follows:

$$\frac{d}{dx} \ln(x) = \frac{1}{x}$$ $$\frac{d}{dx} e^x = e^x$$ $$\frac{d}{dx} e^{ax} = a e^{ax}$$ $$\frac{d}{dx} a^x = a^x \ln(a)$$

When differentiating more complex expressions involving exponentials and logarithms, the chain rule is often required.

Similarly, for logarithmic functions, the logarithmic differentiation technique can be useful when dealing with products, quotients, or exponentials with variable bases.

Example