Compare quantities on a log scale

How should the ratio A/B be found from these base-10 logs, using the same reference unit?

Comparing quantities on a log scale

A base-10 log scale records multiplicative size using exponents, which keeps quantities spanning several powers of ten in a readable range. To compare two original quantities, subtract their log values and then apply the corresponding power of 10.

1. 1Keep the requested order of comparison For A/B, subtract the logarithm of B from the logarithm of A. Do not automatically put the larger logged value first: the requested difference can be negative \(\displaystyle \log_{10}(A/u)-\log_{10}(B/u)=\log_{10}(A/B)\)
2. 2Raise 10 to the difference The difference is an exponent, so the factor between the original quantities is 10 raised to that difference \(\displaystyle 10^3=1000\)
3. 3Interpret one log unit as a factor of ten A one-unit increase on a base-10 log scale multiplies the underlying quantity by 10. \(\displaystyle \log_{10}(A)-\log_{10}(B)=1\Rightarrow\frac{A}{B}=10\)
4. 4Use a log scale for wide ranges Values separated by several powers of ten occupy a shorter, readable range when their logarithms are displayed \(\displaystyle 10^0,10^1,10^2,10^3,10^4,10^5,10^6\Rightarrow0,1,2,3,4,5,6\)

Positive quantities using the same reference unit

\(\displaystyle \log_{10}(A/u)=4,\quad\log_{10}(B/u)=1\)

Find the signed log difference for the ratio A/B: subtract the logarithm of B from the logarithm of A

Positive quantities using the same reference unit

\(\displaystyle \log_{10}(A/u)=4,\quad\log_{10}(B/u)=1\)

For the ratio A/B, raise 10 to the log difference shown to find the comparison factor

Positive quantities using the same reference unit

\(\displaystyle \log_{10}(A/u)=4,\quad\log_{10}(B/u)=1\)

Which line correctly converts the given log values into the ratio A/B?

State what the logarithm measures

Use positive values of the same quantity in the same unit. A precise dimensionless label is the base-10 logarithm of the quantity divided by the common reference unit. A log axis can show equally spaced powers of ten in the original units; an axis explicitly labelled with the base-10 logarithm of the quantity divided by that reference unit instead shows their exponents

1. 1Equal intervals mean equal factors The positions of 1, 10, 100 and 1000 in the same unit are equally spaced. Equal arithmetic increases are generally not equally spaced
2. 2Respect the order of comparison For the ratio of A to B, subtract the logarithm of B from the logarithm of A, even if the difference is negative. Reverse the order to obtain the reciprocal ratio
3. 3Distinguish model predictions from observations When a task states that displayed logs are exact model predictions, calculate from those exact logs and round only the requested answer. A modelled expected count is a predicted mean across repeated comparable cases and may be fractional; it is not an observed whole-number count. Rounded logarithms of observations instead support approximate reconstructions, not an unstated exact ratio

Data or rule

\(\displaystyle \log_{10}(A/u)=1,\quad\log_{10}(B/u)=3\)

Both logs use the same reference unit. Find A/B in the stated order

Data or rule

\(\displaystyle \log_{10}(A/u)=1,\quad\log_{10}(B/u)=3\)

Find the reversed ratio B/A for those same measurements

The values 10, 20 and 30 share a common unit. Are their positions equally spaced on a base-10 log axis?

Data or rule

\(\displaystyle \log_{10}(1)=0,\quad\log_{10}(100)=2\)

What positive value lies halfway between 1 and 100 on a base-10 log axis, in the same unit?

One mass is logged in grams and another in kilograms. What must happen before subtracting their log values to compare masses?

Construct a log-scale representation

Four positive measurements are 0.1, 1, 10 and 100 metres. Draw a labelled base-10 log axis with equally spaced decade ticks, place each measurement, and also state its coordinate on an axis labelled with the base-10 logarithm of the measurement divided by one metre. Add the point at the square root of ten metres. Compare the largest with the smallest, explain the spacing and say whether zero metres can be plotted

Model response: Construct a log-scale representation

1. 1Coordinates and labels The four exponent coordinates are −1, 0, 1 and 2, so equally spaced ticks can be labelled 0.1, 1, 10 and 100 m. Alternatively label the axis with the base-10 logarithm of the measurement divided by one metre, with ticks −1, 0, 1 and 2. Do not label the exponent values as metres
2. 2Intermediate point and ratio The square root of ten metres has exponent one-half, halfway between 1 m and 10 m. The log difference from 0.1 m to 100 m is three, so the ratio is ten cubed, or 1000. Each equal interval multiplies the original measurement by ten
3. 3Domain Zero has no finite real base-10 logarithm and cannot occupy a point on this axis. Positive values below the unit have negative exponent coordinates, not negative original measurements