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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

## Logarithm Solver Log(?)=x

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

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

### 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?

— $\log_b(b) = 1$

— $\log(e) = \ln(e) = 1$

— $\log_{10}(10) = 1$

— $\log_b(1) = ln(1) = 0$

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

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