Logarithmic function

In the previous chapter we saw that the exponential $\exp_a(x) = a^x$ is a bijective function from $\mathbb{R}$ to $(0;+\infty)$. By the theorem on the existence of the inverse we can then define its inverse function: it is precisely the logarithm. This chapter introduces the definition of logarithm as an exponent, the fundamental properties (product, quotient, power) and the change-of-base formula, studies the graph of the logarithm as the reflection of that of the exponential, and develops the methods for solving logarithmic equations and inequalities. It closes with a broad section of applications on logarithmic scales — decibels, Richter scale, pH, stellar magnitudes — where the logarithm shows its usefulness in describing phenomena that vary over many orders of magnitude.

Sections

• Definition
• Properties of logarithms
• Change of base
• Graph and inverse function
• Logarithmic equations
• Logarithmic inequalities
• Logarithmic scales and decibels

Exercises

• All the worked exercises of the chapter