Semi-logarithmic paper

Statement

On semi-logarithmic paper ($y$ axis logarithmic, $x$ axis linear) plot $y = 3\cdot 2^x$ for $x\in[0,5]$. Verify that the graph is a straight line and compute its slope in the system $(x, \log_{10} y)$.

Solution

Applying $\log_{10}$ to both sides: $\log_{10} y = \log_{10}(3\cdot 2^x) = \log_{10} 3 + x\log_{10} 2.$ In the plane $(x,\ \log_{10} y)$ this is the equation of a straight line with intercept $\log_{10} 3 \approx 0{,}477$ and slope $\boxed{\log_{10} 2 \approx 0{,}301}.$ The values $y = 3, 6, 12, 24, 48, 96$ (for $x = 0,1,2,3,4,5$), plotted on a logarithmic $y$ axis, indeed line up.

On a logarithmic $y$ axis, the exponential $y=3\cdot 2^x$ becomes a straight line: this is the point of semi-logarithmic paper.

Links

Topics: Logarithmic function
Concepts: Graph of a function · Logarithm · Logarithmic scale
Functions: Exponential function · Logarithmic function
Methods: Logarithmic scale
Skills: Interpret graphs · Draw graphs
Exercise type: Reading a graph