Change of base rule

In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number. In simple cases the logarithm counts factors in multiplication. For example, the base $10$ logarithm of $1000$ is $3$ , as $10$ to the power $3$ is $1000$ ( $1000 = 10 \times 10 \times 10 = 10 3$ ); 10 is used as a factor three times. More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm can be calculated for any two positive real numbers $b$ and $x$ where $b$ is not equal to $1$ . The logarithm of $x$ to base $b$ , denoted $log b (x)$ , is the unique real number $y$ such that $b y = x$ . For example, $log 2 (64) = 6$ , as $64 = 2 6$ .

The logarithm to base $10$ (that is $b = 10$ ) is called the common logarithm and has many applications in science and engineering. The natural logarithm has the number $e$ ( $\approx 2.718$ ) as its base; its use is widespread in mathematics and physics, because of its simpler derivative. The binary logarithm uses base $2$ (that is $b = 2$ ) and is commonly used in computer science.

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