Logarithmic Scale

Definition

Logarithmic Scale

In a Cartesian coordinate system, sometimes, one or more axes are log-scaled, meaning that the points are logarithmically aligned.

Values

Negative Values

Since ${\log }_b(x)$ is not defined for negative values, log-scaled axes can only represent positive values.

$10^0$, which is equal to 1, is the $0$-line which is why negative values cannot be $\le 0$.

Values between 0 and 1

However, values between $(0,1)$ (0 and 1 excluded) are represented in the $0$-line:

Values between Integers

Example: The difference $\log (0.5)-\log (0.4)=0.0969$ is smaller than $\log (0.4)-\log (0.3)=0.1249$. Therefore, the log-values are not equidistant.

Log-Scale for Y:

Log-Scale for X:

Types

X Axis only

Logarithm functions appear as a straight line in a $x$-log coordinate system.

Y Axis only

In a $y$-log coordinate system, exponential functions are represented as straight lines.

Why is it a Straight Line?

Applying the logarithm to an exponential function $f(x)$:

$$\begin{array}{rl}& \\ f(x)& =f_0\cdot e^{\lambda \cdot x}\\ \log (f(x))& =\log (f_0\cdot e^{\lambda \cdot x})\\ \log (f(x))& =\log (f_0)+\log (e^{\lambda \cdot x})\\ \log (f(x))& =\log (f_0)+\lambda \cdot x\cdot \log (e)\end{array}$$

Let $d=\log f_0$ and $k=\lambda \cdot \log (e)$, the equation can be formulated as:

$$\log (f(x))=k\cdot x+d$$

Which is a linear function. Therefore, $y$-logscaled exponential functions are linear functions.

Both axes: X and Y

Power functions appear as a straight line in $xy$-logscaled coordinate systems.