Natural logarithm

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red: natural logarithm

The natural logarithm, ln(x) is the inverse of the function ex. In other words, if y = ex, we define ln(y) = x.

The natural logarithm has some interesting properties that follow from the multiplicative properties of ex. The natural logarithm is also particularly useful in calculating interest.

Properties of the Logarithm

  • ln(ab) = ln(a) + ln(b) for all positive reals a,b.


Proof: If a,b > 0, we can write a = ex and b = ey. It follows that ln(ab) = ln(exey) = ln(ex + y). By definition, ln(ex + y) = x + y = ln(ex) + ln(ey). This last expression, of course, is ln(a) + ln(b).


  • ln(xp) = pln(x)


Proof: If p is a positive integer, this just follows from repeated application of the above-mentioned additive property of the logarithm. For p = \frac{1}{n}, note that the statement follows by observing that ln(x) = ln((x1 / n)n) = nln(x1 / n). Thus, pln(x) = ln(xp) for all rational numbers p. The statement must therefore hold for all reals p by continuity.


  • ln(1) = 0


Proof: \ln(1) = \ln(1\cdot 1) = \ln(1)+\ln(1), whence we must have ln(1) = 0.

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