Logarithms with different bases

Compare three logarithms with different bases: $y = \log_2 x$, $y = \ln x$ and $y = \log_{10} x$. Each is defined for $x > 0$ and equals $0$ at $x = 1$, so all pass through $(1, 0)$. Whatever the base, raising it to the power $0$ gives $1$, so $\log_a 1 = 0$ always holds.

Where each reaches $y = 1$

Each reaches $y = 1$ when $x$ equals its base.

The smaller the base, the faster the growth: $\log_2 x$ gains $1$ every time $x$ doubles, while $\log_{10} x$ needs a tenfold increase for the same gain.

The shape is identical

In truth the three graphs have exactly the same shape, because the change-of-base formula writes every logarithm as a constant multiple of $\ln x$.

Since $\log_2 x \approx 1.44 \ln x$ and $\log_{10} x \approx 0.434 \ln x$, the three curves are one curve stretched or compressed vertically. At any $x$ whatsoever, the three values stand in the fixed ratio $1.44 : 1 : 0.434$.

Properties they share

Their properties therefore coincide. All increase monotonically, all diverge to $-\infty$ as $x \to 0^{+}$, all take the $y$-axis as a vertical asymptote, and all pass through $(1, 0)$. Only the vertical scale differs. A base smaller than $1$ flips things over: $\log_{1/2} x = -\log_2 x$ decreases instead.

Choosing a base

The large dots mark the common point $(1, 0)$ and the points $(2, 1)$ and $(e, 1)$ where $\log_2 x$ and $\ln x$ reach $y = 1$.