Identity

$lo g (ab) = lo g a + lo g b$ and $lo g \frac{a}{b} = lo g a - lo g b$

Algebra • Logarithm • Laws of Logarithms

Formula

lo g (ab) = lo g a + lo g b, lo g \frac{a}{b} = lo g a - lo g b

For

a > 0

b > 0

and any valid base:

lo g (ab) = lo g a + lo g b

and

lo g \frac{a}{b} = lo g a - lo g b

How to use it

1Break a compound argument into tabulated pieces: $lo g_{10} \frac{10}{3} = 1 - lo g_{10} 3$ .
2Run it in reverse to collapse several logarithms of the same base into one, which is what lets two sides be equated.
3Combine with the power rule to flatten any product of powers into a sum of multiples.

Given

lo g_{10} 2 = 0.3010

and

lo g_{10} 3 = 0.4771

, find

lo g_{10} 1.5

Step-by-step

1
$lo g_{10} 1.5 = lo g_{10} \frac{3}{2} = lo g_{10} 3 - lo g_{10} 2$
Why: Split the quotient.
2
$= 0.4771 - 0.3010 = 0.1761$

Answer

0.1761

Watch out

There is no law for a sum inside the logarithm:

lo g (a + b)

cannot be split at all, and it is not

lo g a + lo g b

. Both arguments must be strictly positive.

Builds on

Also known as

log laws, product rule for logarithms, quotient rule for logarithms, split a logarithm, logarithm of a product, logarithm of a quotient