Natural logarithm

red: natural logarithm. Note that the legend is in error in that it confuses the base 2 and base e lines.

The natural logarithm, $\text{[math]}$ is the inverse of the function $\text{[math]}$ . In other words, if $\text{[math]}$ , we define $\text{[math]}$ .

The natural logarithm has some interesting properties that follow from the multiplicative properties of $\text{[math]}$ . The natural logarithm is also particularly useful in calculating interest.

Properties of the Logarithm

$\text{[math]}$ for all positive reals $\text{[math]}$ .

Proof: If $\text{[math]}$ , we can write $\text{[math]}$ and $\text{[math]}$ . It follows that $\text{[math]}$ . By definition, $\text{[math]}$ . This last expression, of course, is $\text{[math]}$ .

$\text{[math]}$

Proof: If $\text{[math]}$ is a positive integer, this just follows from repeated application of the above-mentioned additive property of the logarithm. For $\text{[math]}$ , note that the statement follows by observing that $\text{[math]}$ . Thus, $\text{[math]}$ for all rational numbers $\text{[math]}$ . The statement must therefore hold for all reals $\text{[math]}$ by continuity.