Logarithmic scale

Parametric equations

A logarithmic scale puts ${\log }_{10}y$ (or $\ln y$) on the vertical axis, so each factor of ten in $y$ is the same vertical step. Earthquake magnitude, decibels, and pH compress a wide range this way. The figure always shows the same curves twice: a linear axis on the left, and a log axis on the right.

An exponential $y=C{e}^{kt}$ is a straight line on the $\ln y$ axis:

Interaction

• Exponential slope a: change $a$, and compare the bend on the linear axis with the fixed slope on the log axis
• Compare: overlay Exponential, Power, and Linear on both axes
• Power p and Linear factor m: set the power and the straight line once those curves are on

The two axes stay side by side. Compare does not create them.

What to notice

• An equal step on the log axis is a factor of ten. On the linear axis the same numbers pile up near the bottom.
• The exponential is the curve that becomes straight. The power and the linear curve stay bent on the log axis.
• The two panels use the same values. What changes is whether an equal step is a difference or a ratio.

Related

• Exponential growth and decay
• Geometric definition of the natural logarithm
• Equiangular spiral

Further reading

• Logarithmic scale (Wikipedia)