2.15 Semi-log Plots

Why use semi-log plots?

A semi-log plot graphs data with a linear x-axis but a logarithmic y-axis. If the underlying data is exponential (y = a·b^x), plotting log(y) versus x produces a straight line. This is because taking the log of both sides gives log(y) = log(a) + x·log(b), a linear equation in x.

Reading slope and intercept

On a semi-log plot of log(y) vs. x, the slope equals log(b), where b is the exponential growth/decay factor, and the y-intercept equals log(a), where a is the initial value. So b = 10^(slope) and a = 10^(intercept). This lets you extract exponential model parameters directly from a straight-line graph.

Identifying exponential data

Semi-log plots are a practical tool for testing whether data is exponential: if plotting log(y) against x yields points that fall approximately on a straight line, the original data follows an exponential model. Curved patterns instead suggest a non-exponential relationship, such as polynomial or logarithmic growth.

Key terms

Semi-log plot

A graph with a linear x-axis and a logarithmic y-axis, used to linearize exponential data.

Why exponential data becomes linear on a semi-log plot

Taking log of y = a·b^x gives log(y) = log(a) + x·log(b), which is linear in x.

Slope on a semi-log plot

Equals log(b), the log of the exponential growth/decay factor.

Y-intercept on a semi-log plot

Equals log(a), the log of the initial value.

Curved semi-log plot

Indicates the original data is NOT exponential; a straight-line log(y) transformation only works for exponential models.

Try an AP-style question

On a semi-log plot (log y vs. x), exponential data appear…

1. parabolic
2. periodic
3. exponential
4. linear