Notations
Decimal numbers : are for example noted 2.5.
Exponents express quantities which can be very large (for example the mass of the Sun in grams) or very small (the mass of a hydrogen atom in grams).
The positive exponent gives the number of zeros:
$$10^3=1,000$$
The negative exponent also gives the number of zeros, but with a decimal point: $10^{-3}=0.001=\frac{1}{1,000}$
The mass of the Sun in grams is written: $2 \times 10^{33}$ which can also be written : 2 followed by 33 zeros.
The mass of a hydrogen atom in grams is written: $1.7 \times 10^{-25}$ which can also be written with 25 zeros followed by 17, and a decimal point after the first 0.
The logarithm of a number: this is the function which gives the exponent of a number:
The decimal logarithm, or “log” of $10^{33}$ is 33
This replaces multiplications by additions (which are easier). It is harder to calculate $0.0001 \times 10,000,000,000$ than to calculate $10^{-4} \times 10^{10}$ Simply add the exponents to find $10^{6}$ or one million.
Logarithms are extremely useful for planet hunters, who are manipulating billions of stars and thousandths of arc-seconds. $log (x y) = log(x) + log (y) $
The logarithm is also extremely useful when representing a diagramme. Representing an axis with a logarithmic scale ? shows what happens for the small values of the parameter.