Graph of the Common Logarithm $y = \log_{10} x$

$y = \log_{10} x$ is the logarithm with base $10$: it gives the power to which $10$ must be raised to obtain $x$. Called the common logarithm, it satisfies $10^{\log_{10} x} = x$ and serves as a ruler for measuring digits in base $10$.

Domain and range

Since $10^y$ is always positive, the domain is $x > 0$. The range is all real numbers, and the values are negative whenever $x$ is smaller than $1$.

Monotonicity and shape

The derivative is as follows.

As $\ln 10 \approx 2.303$ is a positive constant, $y' > 0$ for $x > 0$ and the function increases monotonically. Its slope at any given $x$ is about $0.434$ times that of $\ln x$, so it climbs even more gently than the natural logarithm. The second derivative is negative, so the curve is concave down throughout.

Asymptotes

As $x \to 0^{+}$ the function diverges to $-\infty$, so the $y$-axis, the line $x = 0$, is a vertical asymptote. As $x \to +\infty$ it diverges to $+\infty$, but at a crawl: raising the value by $1$ takes a tenfold increase in $x$.

A tenfold increase adds one

The identity $\log_{10}(10x) = \log_{10} x + 1$ holds.

This is also why a positive integer $N$ has $\lfloor \log_{10} N \rfloor + 1$ digits.

Notable points

The curve passes through $(1, 0)$, $(10, 1)$, $(100, 2)$ and $\left( \dfrac{1}{10}, -1 \right)$. With $\log_{10} 2 \approx 0.301$ and $\log_{10} 3 \approx 0.477$, the familiar fact that $2^{10} = 1024$ is close to $10^3$ shows up as $10 \times 0.301 = 3.01$.

Relation to the natural logarithm

The inverse is $10^x$, and the two graphs are reflections of each other in the line $y = x$. The relation to the natural logarithm is as follows.

The graph is that of $\ln x$ compressed vertically by a factor of about $0.434$. The shape is identical, it still passes through $(1, 0)$, and the $y$-axis is still its asymptote: changing the base of a logarithm only rescales the vertical axis.

Applications

It measures orders of magnitude.

One step on such a scale means a fixed multiplicative factor: one more point of earthquake magnitude means roughly $32$ times the energy. The same function underlies semi-log and log-log plots, which fit quantities differing by many orders of magnitude onto a single chart.