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Logarithm

Tool for calculating logarithms with the logarithm function is denoted log or ln, 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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Answers to Questions (FAQ)

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. The dCode calculator always uses $ \log = \ln $.

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: $$ \log_{N}(x) = \frac {\ln(x)} {\ln(N)} $$

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

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 logarithm 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_{10}(10) = 1 $

— $ \log_b(1) = ln(1) = 0 $

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

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Logarithm on dCode.fr [online website], retrieved on 2024-06-24, https://www.dcode.fr/logarithm

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