Logarithmic Function

A logarithmic function is any function where the variable appears inside the argument of a logarithm: $$ y = \log_b x $$ Here $b$ is a positive real number ($b>0$), different from 1 ($b\neq1$), and is called the base of the logarithm.

For every base $b>0$ with $b\neq1$, the logarithmic function is defined only for positive inputs.

Thus, its domain is the set of positive real numbers $R^+$, while its range is the entire set of real numbers $R$.

$$ f : R^+ \rightarrow R $$

The figure below shows the graph of the base-10 logarithm ($\log_{10}$) alongside the natural logarithm ($\ln$), which uses Euler’s number $e$ as its base.

If the base is greater than 1, the logarithmic function is increasing:

• It takes negative values for $0 < x < 1$.
• It takes positive values for $x > 1$.

If the base lies between 0 and 1, the logarithmic function is decreasing:

• It is negative for $x > 1$.
• It is positive for $0 < x < 1$.

In either case, the logarithmic function is bijective and therefore invertible.

The inverse of a logarithmic function is an exponential function of the form $b^y$, with $y = \log_b x$.

Example. The base-2 logarithmic function is invertible. Its inverse is the exponential function $2^y$.

The graphs of $b^y$ and $y = \log_b x$ are symmetric with respect to the line $y = x$, the bisector of the first and third quadrants.

Further Remarks

Some useful facts:

• The logarithm with reciprocal base equals the negative of the logarithm with the original base: $$ \log_{\tfrac{1}{b}} x = - \log_b x $$

  Proof. Consider $\log_{\tfrac{1}{2}} x$. By the change-of-base formula, $$ \log_{\tfrac{1}{2}} x = \frac{\log_2 x}{\log_2 \tfrac{1}{2}} .$$ Since $\log_2 \tfrac{1}{2} = -1$, it follows that $$ \log_{\tfrac{1}{2}} x = \frac{\log_2 x}{-1} = -\log_2 x .$$ This establishes the relationship between a logarithm with base $b$ and one with base $1/b$.

• A logarithmic function symmetric with respect to the $x$-axis can be obtained either by taking the reciprocal base $1/b$ or by simply negating the logarithm, $-\log_b x$, with the same base.
  

And so on…