Logarithmic Scales

Logarithmic scales are scales where intervals increase exponentially
- A normal scale might go 1, 2, 3, 4, ...
- A logarithmic scale might go 1, 10, 100, 1000, ...
Sometimes we can keep the scales with constant intervals by changing the variables
- If the values of x increase exponentially: 1, 10, 100, 1000, ...
- Then you can use the variable log x instead which will have the scale: 1, 2, 3, 4, ...
- This will change the shape of the graph
  - If the graph transforms to a straight line then it is easier to analyse
Any base can be used for logarithmic scales
- The most common bases are 10 and e

For variables that have a large range it can be difficult to plot on one graph

If we are interested in the rate of growth of a variable rather than the actual values then a logarithmic scale is useful

log-log & semi-log Graphs

A log-log graph is used when both scales of the original graph are logarithmic
- You transform both variables by taking logarithms of the values
log y & log x will be used instead of y & x
Power graphs ( $y = a x^{b}$ ) look like straight lines on log-log graphs

A semi-log graph is used when only one scale (the y-axis) of the original graph is logarithmic
- You transform only the y-variable by taking logarithms of those values
log y will be used instead of y
Exponential graphs ( $y = a b^{x}$ ) look like straight lines on semi-log graphs

Identify whether one or both of the scales are logarithmic
Identify the variable so that the scales have equal intervals
- x : 1, 10, 100, 1000, ... use log x
- For x : 1, e, e², e³, ... use ln x
If you are asked to estimate a value:
- First find the value of any logarithms
  - For example: log y, ln x, etc
- Use the graph to read off the value
- If it is a value for a logarithm find the actual value using:
  - $\log x = k \Rightarrow x = 10^{k}$
  - $\ln x = k \Rightarrow x = e^{k}$

The function $y = f (x)$ is drawn below using a log-log graph.

2-7-1-we-image Show that when $x = 56$ the value of $y$ is approximately 24.

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