Logarithm

The definition of the natural logarithm is the function whose derivative is the inverse function of $ x \mapsto \frac 1 x $ defined for $ x \in \mathbb{R}_+^* $.

The natural logarithm is noted log or ln and is based on the number $ e \approx 2.71828\ldots $ (see decimals of number e).

Any base $ N $ logarithm can be calculated from a natural logarithm with the formula: $$ \log_{N}(x) = \frac {\ln(x)} {\ln(N)} $$

It follows that $ log_{a}(b) = \frac {\ln(b)} {\ln(a)} $ and $ log_{b}(a) = \frac {\ln(a) } {\ln(b)} $ are inverses

The neperian logarithm is the other name of the natural logarithm (with base e).

The decimal logarithm noted $ \log_{10} $ or log10 is the base $ 10 $ logarithm. This is one of the most used logarithms in calculations and logarithmic scales. $$ \log_{10}(x) = \frac {\ln(x)} {\ln(10)} $$

The binary logarithm noted $ \log_{2} $ (or sometimes $ lb $) is the base $ 2 $ logarithm. This logarithm is used primarily for computer calculations. $$ \log_2(x) = \frac {\ln(x)} {\ln(2)} $$

Any logarithm has as for properties:

— $ \log_b(x \cdot y) = \log_b(x) +\log_b(y) $ (transformation of a product into a sum)

— $ \log_b \left( \frac{x}{y} \right) = \log_b(x) - \log_b(y) $ (transformation of a quotient into subtraction)

— $ \log_b (x^a) = a \log_b(x) $ (transformation of a power into a multiplication)

— $ \log_b(b) = 1 $

— $ \log(e) = \ln(e) = 1 $

— $ \log_{10}(10) = 1 $

— $ \log_b(1) = ln(1) = 0 $

— $ \log_b(b^n) = \ln(e^n) = n $ (inverse function of exponentiation)