Logarithms

You know that $2^3 = 8$. But what if the question runs the other way: with which exponent does $2$ turn into $8$? This reversal of taking a power is the logarithm. It answers exactly the questions that are often the more interesting ones in everyday life: how many doublings lie between $1$ and $1024$? Across how many powers of ten are we talking here?

The reverse question

The logarithm $\log_b(x)$ asks: with which exponent does the base $b$ turn into the number $x$? Because $2^3 = 8$, we have $\log_2(8) = 3$. The base $b$ sits small below, the number $x$ in the bracket, and out comes the exponent we seek. The base is always positive and never $1$, and $x$ has to be greater than $0$: a power of a positive base is never $0$ and never negative. Taking a power and taking a logarithm are two sides of the same relation, they undo each other.

The laws of logarithms

Logarithms turn multiplication into addition, that was once their purpose in calculation. A product inside the bracket becomes the sum of two logarithms: $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$. A power inside the bracket becomes the factor in front: $\log_b(x^n) = n \cdot \log_b(x)$. So a big multiplication shrinks to a comfortable addition.

The base ten logarithm

The base $10$, the common logarithm, is used especially often. For powers of ten it simply counts the zeros: $\log_{10}(1000) = 3$, because $1000$ has three zeros and $10^3 = 1000$. Scales like earthquake magnitude or the pH value rest on exactly this: one step on the scale stands for a factor of $10$, for an earthquake ten times the deflection on the seismometer. The logarithm makes such huge spans manageable again.