Exponential and logarithmic functions

Exponential and logarithmic functions are inverses of each other. Therefore, their graphs are reflections of each other across $y = x$ diagonal line.

Contents

• $\e$
• Properties
• Shapes
  • shapes of Exponential functions
  • shapes of Logarithmic functions
• Monotonicity

$\e$

$f(x) = \e^x$ is a special function where the tangent line at $\left(x_1, \e^{x_1}\right)$ has a gradient of $\e^{x_1}$. In calculus lingo,

Logarithms with $\e$ as the base are called the natural logarithm ($\ln$). YouTube: Natural logs were tabulated before the discovery of the number $\e$. 

Properties

See also exponent and log rules

For any arithmetic sequence $u_n$, it is known that $k^{u_n}$ is a geometric sequence, for any $k > 0$. See definitions of arithmetic and geometric sequences.

Exponential growth is faster than any polynomial growth for sufficiently large inputs.

Logarithmic growth is slower than any $\sqrt[n]{x}$, for sufficiently large $x$.

Shapes

Unlike functions covered up to this point, exponential and logarithmic functions lack symmetry in that with and without vertical reflection and with and without horizontal reflection results in four different shapes. $x\mapsto x^2$, for instance, appears the same whether there is a horizontal reflection. And applying either horizontal reflection or vertical reflection on $x\mapsto x$ produces the same effect.

The following discusses this in more depth, and links to desmos interactive graphs that you can use to explore the shapes.

shapes of Exponential functions

Desmos playground for exponential functions 

Compared to $e^x$, $A$ combines the vertical stretch (and/or reflection), and the horizontal shift, while $b^c$ collectively controls the horizontal stretch (and/or reflection).

The horizontal asymptote is $y=k$.

If $A > 0$, the function is above the asymptote; if $A < 0$, function is below.

If $b^c > 1$, the graph diverges from asymptote on the right side; if $b^c < 1$, the graph diverges from asymptote on the left side.

shapes of Logarithmic functions

Desmos playground for logarithmic functions 

Compared to $\ln x$, $c$ combines the horizontal stretch (and/or reflection), and the vertical shift, while $b^A$ collectively controls the vertical stretch (and/or reflection).

The vertical asymptote is $x=h$.

If $c > 0$, the function is to the right of the asymptote; if $c < 0$, it is to the left.

If $b^A > 1$ the graph diverges from asymptote on the top side; if the result is $b^A < 1$, the graph diverges from asymptote on the bottom side.

Monotonicity

All exponential and logarithmic are either strictly increasing or strictly decreasing.

For a strictly increasing function, eg $f(x) = \ln x$, we have

and for a strictly decreasing function, eg $g(x) = -\ln x$, we have

assuming that $a, b$ are both in the domain of the particular function.