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Logarithm

Tool for calculating logarithms. The logarithm function is denoted log or ln and is defined by a base (the base e for the natural logarithm).

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Logarithm -

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Logarithm

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Tool for calculating logarithms. The logarithm function is denoted log or ln and is defined by a base (the base e for the natural logarithm).

Answers to Questions

What is the natural logarithm? (Definition)

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).

Example: $ \log(7) = \ln(7) \approx 1.94591 $

log

Some people and bad calculators use $ \log $ for $ \log_{10} $, so make sure to know which notation is used.

How to turn a base N logarithm into a natural logarithm?

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

What is the neperian logarithm?

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

What is the decimal logarithm (log10)?

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. $$ \log_{10}(x) = \frac { \ln(x)} { \ln(10) } $$

Example: $ \log_{10}(1000) = 3 $

What is the binary logarithm (log2)?

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

Use the formula above to calculate a log2 with a calculator with only the log key.

Why the logaritm can transform product into sum?

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)

What are remarkable values of the logarithm function?

- $ \log_b(b) = 1 $

- $ \log(e) = \ln(e) = 1 $

- $ \log_b(1) = ln(1) = 0 $

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

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