Exponential and logarithm

Idea

For $a>0$ and $a\ne 1$, the exponential $y=a^x$ and the logarithm $y={\log }_{a}x$ are inverses of each other. The base $e\approx 2.718$ is the one continuous compounding and calculus use.

This page keeps that relationship and only changes the reading. First read it as a mirror across $y=x$. Then put the vertical axis on a log scale, so doubling becomes a constant slope. Then use area to see why $e$ is natural. One model, in more detail, is left to the single works linked below.

Interaction

• Exponential and logarithm: move Base a and Coefficient b. On Linear coordinates, $y=b\cdot a^x$ and $y={\log }_a(x/b)$ mirror each other across $y=x$. Switch the axis button to log y coordinates and that same exponential becomes a straight line of slope $\ln a$
• Birth of e: raise Partitions n and watch $(1+1/n{)}^n$ approach $e$. Move Area endpoint x until the area under $y=1/x$ is $1$
• Change of base: change Base a. Every ${\log }_{a}x$ passes through $(1,0)$, and the base only rescales the graph through ${\log }_{a}x=\ln x/\ln a$

Suggested order: Exponential and logarithm, then Birth of e, then Change of base.

What to notice

• The exponential and the logarithm are symmetric across $y=x$. On log y coordinates, doubling on that same curve becomes a constant slope.
• Changing the axis does not change the values. It changes what an equal step means: a fixed difference on a linear axis, and a fixed ratio on a log axis.
• The mirror and the log axis are still the shape of the same exponential. What is special about $e$ is which base makes a unit area match a unit logarithm, not a more curved graph.

Related

• Exponential growth and decay
• Logarithmic scale
• Geometric definition of the natural logarithm
• Logistic curve

Further reading

• Natural logarithm (Wikipedia)
• Logarithmic scale (Wikipedia)