IB Maths AI 1.8 Notes

Logarithms

A logarithm is the inverse of an exponential. If $a^x=b$, then

$\log_ab=x$

This means a logarithm tells us the power that the base must be raised to in order to produce a number.

Example: $2^3=8\iff\log_28=3$.

A logarithm with base $10$ is called a common logarithm.

• $\log1000=3$ because $10^3=1000$
• $\log0.001=-3$ because $10^{-3}=0.001$

A logarithm with base $e$ is called a natural logarithm and is written as $\ln$.

$\ln x=\log_ex$

The number $e$ is an irrational constant with value approximately $2.71828$.

Important facts for $a \gt 0$, $a\ne1$, and $x \gt 0$:

• $\log_aa=1$
• $\log_a1=0$
• $\log_a(a^x)=x$
• $a^{\log_ax}=x$
• $\ln(e^x)=x$
• $e^{\ln x}=x$