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Arithmetic with Logarithms

Arithmetic with logarithms can seem pretty strange at first. Unlike most of the other things that we talk about doing arithmetic with in algebra, with logarithms the interesting things happen with addition instead of multiplication. In fact, the logarithm rules establish some strange relationships between addition/subtraction and multiplication/division. Take a look at addition first.

log(x) + log(y) = log(xy)

In English, this says that the sum of the logarithms of two numbers is equal to the logarithm of the product of the numbers. Once you've swallowed that one, the subtraction formula goes down pretty easily.

log(x) - log(y) = log(x / y)

I stated both of the formulas using the default base 10 logarithm but they work just as well for any base.

Some Other Formulas

Logarithms have some other formulas that are useful when solving equations.

log(xy) = y log(x)
log(1) = 0
log(1 / x) = -log(x)

Again, even though I stated the formulas using base 10, they work for any base. The second equation is a direct result of the definition of 0 exponent, i.e. x0 = 1 for any x ≠ 0. The third equation comes from the first one combined with the definition of a negative exponent, i.e. 1 / x = x-1.