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 $(1, 0)$.

Each reaches $y = 1$ when $x$ equals its base: $\log_2 x$ at $x = 2$, $\ln x$ at $x = e \approx 2.72$, and $\log_{10} x$ at $x = 10$. The smaller the base, the faster it grows. The large dots mark the common point $(1, 0)$ and the points $(2, 1)$, $(e, 1)$ where $\log_2 x$ and $\ln x$ reach $y = 1$.