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 $y = 2^x$.

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

Listing the points it passes through shows that the value gains exactly $1$ each time $x$ doubles.

• $(1, 0)$: $\log_2 1 = 0$
• $(2, 1)$: $\log_2 2 = 1$
• $(4, 2)$: $\log_2 4 = 2$
• $(8, 3)$: $\log_2 8 = 3$
• $\left(\dfrac{1}{2},\, -1\right)$: $\log_2 \dfrac{1}{2} = -1$

As $x \to 0^{+}$ it diverges to $-\infty$, so the $y$-axis ($x = 0$) is a vertical asymptote; as $x \to +\infty$ it keeps rising slowly to $+\infty$. The derivative $y' = \dfrac{1}{x \ln 2}$ is always positive and the second derivative is negative, so the graph is concave.

By the change-of-base formula, $\log_2 x = \dfrac{\ln x}{\ln 2} \approx 1.4427\,\ln x$, merely a constant multiple of the natural logarithm. Thus it differs from $\ln x$ or $\log_{10} x$ only by a vertical stretch and has the same shape.

The base-2 logarithm is especially important in computer science. The unit of information, the bit, is measured with $\log_2$: distinguishing $n$ alternatives requires $\log_2 n$ bits. Algorithms that repeatedly halve a problem, such as binary search and merge sort, run in $O(\log_2 n)$ time, and it also appears when measuring a musical octave (a doubling of frequency).