Logarithmic and exponential functions

The natural exponential function is $f(x) = e^x$. Its graph passes through $(0, 1)$ and increases at a rate proportional to its current value. It is always positive. The natural logarithmic function, $g(x) = \ln x$, is defined as the inverse of $e^x$. This means that if $y = e^x$, then $x = \ln y$. The function $\ln x$ is only defined for positive values of $x$, and its graph passes through $(1, 0)$.

The laws of logarithms, which you may have met for base 10, apply equally to natural logarithms. These rules are essential for manipulating and solving equations involving logs and exponentials. Mastering them is key to simplifying complex expressions before differentiation or integration.

For $x, y > 0$:

1. Product Rule: $\ln(xy) = \ln x + \ln y$
2. Quotient Rule: $\ln\left(\frac{x}{y}\right) = \ln x - \ln y$
3. Power Rule: $\ln(x^n) = n \ln x$

One of the most remarkable properties of the function $y=e^x$ is that it is its own derivative. The derivative of $\ln x$ is $1/x$. For more complex functions, we must apply the chain rule.

Standard derivatives: $\frac{d}{dx}(e^x) = e^x$ $\frac{d}{dx}(\ln x) = \frac{1}{x}$

Using the chain rule: $\frac{d}{dx}(e^{f(x)}) = f'(x)e^{f(x)}$ $\frac{d}{dx}(\ln(f(x))) = \frac{f'(x)}{f(x)}$

Exponential functions are used to model many real-world phenomena where the rate of change of a quantity is proportional to the quantity itself. This includes population growth, radioactive decay, and compound interest. The general form is $N = N_0 e^{kt}$, where $N_0$ is the initial value, $t$ is time, and $k$ is a constant determining the rate of growth ($k>0$) or decay ($k<0$).