SL 1.5—Intro to logs Notes

Logarithms: An Introduction

1. Logarithms are one of the inverse functions to exponentiation, the other being radicals.
2. Let's say we have $a$, $b$, and $c$ such that $a^b = c$.
3. To rearrange this equation for $a$, we can use a radical, making $a = \sqrt[b]{c}$.
4. But if we want to rearrange for $b$ instead, we have to use a logarithm.
5. This equation looks like: $$b = \log_a c$$
6. So essentially, if you have $\log_a c$, it's asking "What number do I have to raise $a$ to in order to make $c$?

Base 10 Logarithms

1. The base 10 logarithm, often written as $\log_{10}$ or simply $\log$, is the most commonly used logarithm in everyday applications.
2. It represents the power to which 10 must be raised to obtain a given number.

Natural Logarithms (Base $e$)

1. The natural logarithm, denoted as $\ln$ or $\log_e$, uses the mathematical constant e (approximately 2.71828) as its base.
2. Natural logarithms are particularly important in calculus and many scientific applications.

Numerical Evaluation of Logarithms

1. In practice, logarithms are often evaluated using technology such as scientific calculators or computer software.
2. This means a GDC when you're taking your exams.
3. This is because exact values of logarithms are often irrational numbers.

Logarithmic graphs

1. As mentioned earlier, the logarithm function is the inverse of the exponential function.
2. Thus, the graph of $\log_a x$ looks like the graph of $a^x$ reflected about the line $y = x$.
3. Some properties of a logarithmic graph are:
   1. Domain of $x > 0$
   2. Range of $x \in \mathbb{R}$
   3. Vertical asymptote at $x = 0$
   4. Undefined for $x \leq 0$
   5. $x$-intercept at $(1, 0)$
   6. Increasing if base is greater than 1 (e.g. $\log_{10}x$, $\ln x$)
   7. Decreasing if base is less than 1 (e.g. $\log_\frac12x$)