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

$\e$

$f(x) = \e^x$ 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

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.