Logarithms: An Introduction
- Logarithms are one of the inverse functions to exponentiation, the other being radicals.
- Let's say we have $a$, $b$, and $c$ such that $a^b = c$.
- To rearrange this equation for $a$, we can use a radical, making $a = \sqrt[b]{c}$.
- But if we want to rearrange for $b$ instead, we have to use a logarithm.
- This equation looks like: $$b = \log_a c$$
- So essentially, if you have $\log_a c$, it's asking "What number do I have to raise $a$ to in order to make $c$?
If $2^3 = 8$, then $\log_2 8 = 3$ If $10^x = 1000$, then $\log_{10} 1000 = x$ (which we know is 3)
Base 10 Logarithms
- The base 10 logarithm, often written as $\log_{10}$ or simply $\log$, is the most commonly used logarithm in everyday applications.
- It represents the power to which 10 must be raised to obtain a given number.
- $\log_{10} 100 = 2$ because $10^2 = 100$
- $\log_{10} 1000 = 3$ because $10^3 = 1000$
Natural Logarithms (Base $e$)
- The natural logarithm, denoted as $\ln$ or $\log_e$, uses the mathematical constant e (approximately 2.71828) as its base.
- Natural logarithms are particularly important in calculus and many scientific applications.
- It is crucial to remember that $\log_e x = \ln x$.
- These notations are equivalent and both refer to the natural logarithm.
Numerical Evaluation of Logarithms
- In practice, logarithms are often evaluated using technology such as scientific calculators or computer software.
