Graph of the Binary Logarithm $y = \log_2 x$

$y = \log_2 x$ is the logarithm with base $2$: it gives the power to which $2$ must be raised to obtain $x$. For example $2^3 = 8$, so $\log_2 8 = 3$. It is the inverse of the exponential $y = 2^x$.

Domain and range

The argument $x$ must be positive, so the domain is $x > 0$ and the range is all real numbers. Since the base exceeds $1$ the function increases monotonically, and it is negative for $0 < x < 1$.

Doubling adds one

Each time $x$ doubles, the value gains exactly $1$.

Monotonicity and asymptote

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 keeps rising slowly to $+\infty$. The derivative is as follows.

It is always positive, and the second derivative is negative, so the graph is concave down. At $(1, 0)$ the tangent has slope $y'(1) = \dfrac{1}{\ln 2} \approx 1.4427$.

Symmetry with the exponential

The graph is symmetric to that of $2^x$ in the line $y = x$: where $2^x$ shoots upward, $\log_2 x$ only creeps.

Change of base

By the change-of-base formula it is a constant multiple of the natural logarithm.

It therefore differs from $\ln x$ or $\log_{10} x$ only by a vertical stretch, and the shape is the same.

Applications

The base-$2$ logarithm is especially important in computer science.

• The bit, the unit of information, is measured with $\log_2$
• Distinguishing $n$ alternatives requires $\log_2 n$ bits
• Binary search and merge sort, which halve the problem at every step, run in $O(\log_2 n)$ time
• A musical octave, a doubling of frequency, is measured the same way

Learning that an event of probability $p$ has occurred carries $-\log_2 p$ bits of information, which is the basis of Shannon entropy.