Change of logarithm base: Definition & Examples

Change of logarithm base rewrites a logarithm using logarithms to another base:

\log_b x=\frac{\ln x}{\ln b},\qquad x>0,\quad b>0,\quad b\ne1.

Here

\ln

is the natural logarithm, with base

e

. A calculator with only natural or base-10 logs can therefore evaluate any admissible real base. More generally,

\log_b x=\frac{\log_a x}{\log_a b},\qquad a>0,\quad a\ne1.

Both logs in the quotient must use the same base. For example,

\log_{10}1000=\ln(1000)/\ln(10)=3

Why the denominator is there

Set $y=\log_b x$ . By definition, $b^y=x$ . Taking natural logs gives $y\ln b=\ln x$ , so division by $\ln b$ yields the formula. The denominator converts the natural-log scale into units of one multiplication by $b$ .

Base 1 is excluded because $1^y$ is always 1: it cannot invert an exponential over positive inputs, and $\ln1=0$ would make the quotient undefined. Bases between zero and one are valid, but their logs decrease as the argument increases. For example, $\log_{1/2}(1/4)=2$ . Zero and negative arguments have no real logarithm; complex logarithms require a different treatment.

A growth clock, with its assumption exposed

Suppose a hypothetical investment starts at USD 100 and follows the exponential function

V(t)=100\times2^t,

where $t$ counts periods. It doubles each period, with exponential interpolation between period boundaries; this is a teaching model, not a return forecast. To reach USD 150, solve $2^t=1.5$ :

t=\log_2(1.5) =\frac{\ln1.5}{\ln2} =\frac{\log_{10}1.5}{\log_{10}2} \approx0.584963\text{ periods}.

Knowing only that wealth doubles at period ends would not determine this fractional-period arrival time. The path assumption supplies that information. If growth is credited only at the ends of whole periods, the first credited balance at least USD 150 occurs at period 1 instead.

Synthetic exponential wealth rises from USD 100 at period 0 to USD 200 at period 1 and reaches USD 150 at 0.584963 periods. Natural-log and base-10-log quotients give the same time.

Original synthetic growth clock. The fractional arrival time assumes exponential growth throughout the period.

A base change is not a return-convention change

For the same positive price relative $150/100=1.5$ , with no cash distributions, these quantities differ:

Quantity	Value	Meaning
Simple price return	$0.5$	A 50% increase in price
Natural-log price return	$\ln1.5\approx0.405465$	The conventional continuously compounded return over the interval
Base-10 log price relative	$\log_{10}1.5\approx0.176091$	The same relative on another logarithmic scale

The base-10 quantity must be multiplied by $\ln10$ to recover the natural-log return. Calling it a continuously compounded return without conversion changes the numerical convention. A price ratio is dimensionless; dividends or other distributions need an explicit total-return treatment.

Similarly, changing a chart's logarithmic axis from base 10 to base 2 rescales its log coordinates. Equal multiplicative ratios still occupy equal distances, but the numerical coordinates change. State the base whenever an unqualified “log” could be ambiguous.

For the surrounding finance applications, Exponentials and Logarithms develops discounting, continuous rates and the addition of log-returns through time (public preview; full lesson requires membership).

Sources

Stewart, J. (2008). Single Variable Calculus: Early Transcendentals, 6th edition, §1.6, “Inverse Functions and Logarithms,” Change of Base Formula 10.
OpenStax, Calculus Volume 1, §1.5: Exponential and Logarithmic Functions, change-of-base identities.

Why the denominator is there

A growth clock, with its assumption exposed

A base change is not a return-convention change

Sources

Related terms

Exponential function

Logarithm

Natural logarithm