Logarithms & Log Scales

A logarithm is the index you needed. log₁₀ 1000 = 3 because 10³ = 1000. The log and the power are the same statement written two ways, which is why the check readout takes the log and raises ten to it, landing back on x.

Logs turn multiplying into adding. log(a × b) = log a + log b, which is the property slide rules were built on, and it is why very large ratios become manageable differences once expressed as logs.

A log scale gives each decade the same width. One to ten occupies the same space as ten to a hundred and as a hundred to a thousand, so a range spanning thousands stays readable. Decibels, frequency response and vibration charts all rely on this.

• Try this: Set x = 10, then 100, then 1000. What happens to log₁₀ x each time, and by how much does it step?
• Try this: Watch the marker on both scales as you sweep x from 1 to 1000. Where does the linear scale become useless?
• Try this: Find x where log₁₀ x = 2.5. What is that number, and is it halfway between 100 and 1000?