The logarithm of a number is the exponent to which a base (usually 10) must be raised to produce that number.
Mathematically, if
$$\log_a(x)=n \iff x=a^n$$
$\log_{10}(1000) = 3$ because $10^3 = 1000$
This property allows us to represent large numbers with smaller, more manageable values:
Consider a dataset of bacterial growth where the population starts at 100 and doubles every hour:
Hour 0: 100
Hour 1: 200
Hour 2: 400
Hour 3: 800 ...
Hour 10: 102,400
Taking the $\log_{10}$ of these values gives:
Hour 0: 2
Hour 1: 2.30
Hour 2: 2.60
Hour 3: 2.90 ...
Hour 10: 5.01
This scaled version is much easier to work with and plot.
When dealing with numbers that span several orders of magnitude, always consider using logarithmic scaling to make the data more manageable.
The natural logarithm (denoted as $\ln(x)$) is a logarithm with base $e$, where $e \approx 2.718$ is the mathematical constant known as Euler’s number.
It is widely used in exponential growth and decay models, finance, physics, and calculus.
The natural logarithm is defined as:
$$\ln(x) = n \iff e^n = x$$
This means that $\ln(x)$ gives the exponent to which $e$ must be raised to obtain $x$.
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