Revise Logarithmic and exponential functions for Cambridge 9709 (2.2)

An exponential function is of the form $y = a^x$, where the base $a$ is a positive constant. A logarithm is the inverse of this operation. The statement $y = a^x$ can be rewritten in logarithmic form as $x = \log_a y$. In simple terms, the logarithm gives you the power you need.

Because they are inverse functions, the graph of $y = \log_a x$ is a reflection of the graph of $y = a^x$ in the line $y=x$. The domain of $y=a^x$ is all real numbers, and its range is $y>0$. For its inverse, $y=\log_a x$, this is swapped: the domain is $x>0$ and the range is all real numbers. This is a critical point: you can only take the logarithm of a positive number.

To manipulate logarithmic expressions and solve equations, we use a set of rules known as the laws of logarithms. These laws are directly derived from the laws of indices, which you should already be familiar with. Mastering these is non-negotiable for success in this topic.

For a base $a > 0$ and $a \neq 1$, and for $x, y > 0$:

1. Product Rule: $\log_a(xy) = \log_a x + \log_a y$
2. Quotient Rule: $\log_a(x/y) = \log_a x - \log_a y$
3. Power Rule: $\log_a(x^n) = n \log_a x$

A special base for exponential and logarithmic functions is the number $e$. It is an irrational number, approximately 2.718, and arises naturally in situations involving continuous growth or decay. The function $y=e^x$ is called the natural exponential function, and its inverse, $y=\log_e x$, is called the natural logarithm, usually written as $y=\ln x$. These are the most common exponential and log functions used in science, engineering, and finance.

The constant $e$: $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.71828$ Natural Logarithm: $\ln x \equiv \log_e x$