# Natural Logarithms

Logarithms with base e, where e is an irrational number whose value is 2.718281828... are called natural logarithms. The natural logarithm of x is denoted by ln x. Natural logarithms are widely used in mathematics, physics and engineering.

## Relationship between natural logarithm of a number and logarithm of the number to base a

Let a be the base of logarithm (a > 0, a ≠ 1), and let

$y = {\log _a}x.$

This yields

${a^y} = x.$

By taking the natural logarithm of both sides, we have

$\ln {a^y} = \ln x,\;\; \Rightarrow y\ln a = \ln x,\;\; \Rightarrow y = \frac{1}{{\ln a}}\ln x,\;\; \Rightarrow {\log _a}x = \frac{{\ln x}}{{\ln a}}.$

The last formula expresses logarithm of a number $$x$$ to base $$a$$ in terms of the natural logarithm of this number. By setting $$x = e,$$ we have

${\log _a}e = \frac{1}{{\ln a}}\ln e = \frac{1}{{\ln a}}.$

If $$a = 10,$$ we obtain:

${\log _{10}}x = \lg x = M\,{\ln x} ,\;\;\; \text{where}\;\;M = \frac{1}{{\ln a}} = \lg e \approx 0.43429 \ldots$

The inverse relationship is

$\ln x = \frac{1}{M}\lg x,\;\;\; \text{where}\;\;\frac{1}{M} = \ln 10 \approx 2.30258 \ldots$

Graphs of the functions $$y = \ln x$$ and $$y = \lg x$$ are shown in Figure $$1.$$

## Solved Problems

### Example 1.

Calculate $\ln {\frac{1}{{\sqrt e }}}.$

Solution.

$\ln \frac{1}{{\sqrt e }} = \ln {e^{ - \frac{1}{2}}} = - \frac{1}{2}\ln e = - \frac{1}{2}.$

### Example 2.

Write as one logarithm:

${\frac{1}{3}}\ln \left( {x - 1} \right) - {{\frac{1}{2}}\ln \left( {x + 1} \right)} + {2\ln x}.$

Solution.

$\frac{1}{3}\ln \left( {x - 1} \right) - \frac{1}{2}\ln \left( {x + 1} \right) + 2\ln x = \ln {\left( {x - 1} \right)^{\frac{1}{3}}} - \ln {\left( {x + 1} \right)^{\frac{1}{2}}} + \ln {x^2} = \ln \sqrt[3]{{x - 1}} - \ln \sqrt {x + 1} + \ln {x^2} = \ln \frac{{{x^2}\sqrt[3]{{x - 1}}}}{{\sqrt {x + 1} }}.$

See more problems on Page 2.