Change of Base and the Natural Log

A calculator only has one or two log buttons, usually ln (base e) and log (base 10). The change of base formula lets you reach any base from there: log base b of v = ln(v) / ln(b). You're not doing anything mysterious, just rewriting the same number in a form a calculator can actually compute. The formula works because ln(v) = ln(b) times log base b of v — that's just the definition of logarithm restated in exponent form — and dividing both sides by ln(b) isolates log base b of v.

This is also why every logarithmic graph is secretly the same curve. Since log base b of v = ln(v)/ln(b), the graph of log base b of x is just y = ln(x) stretched or squeezed vertically by the constant 1/ln(b). Change the base, and you're not changing the curve's shape or its domain — you're only changing how tall it stands at each x-value. As b grows from 2 up toward 10, ln(b) grows too, so 1/ln(b) shrinks, and the graph gets flatter: it still climbs forever and still has a vertical asymptote at x = 0, just more slowly.

This connects directly to comparing logs with different bases: since log base b of v = ln(v)/ln(b), a bigger base always produces a smaller log value for the same v, because you're dividing by a bigger number. It also sets up later work modeling data, where you'll need to convert between bases to compare growth rates or fit a log model using whatever base your calculator or software prefers.