Logarithmic: logax = y
Exponential: ay = x
Common logs have a base of 10. (ex. log10x). If a log is written without a base (as log x), then it is assumed to have a base of 10.
Natural logs: These are logs with a base of e. e is a mathematical constant that is equal to the limit of (1 + 1/n)n as n approaches infinity, approximately 2.718281828. (It has many more digits than those written here.) logex is often written as ln x.
loga(xy) = logax + logay
A log of two numbers, x and y, that are being multiplied by each other can be split into two separate logs: a log of each of the factors being added together. (This also works in reverse.)
log28 + log22
loga(x/y) = logax - logay
A log of a two numbers being divided by each other, x and y, can be split into two logs: the log of the dividend x minus the log of the divisor y.
log25 - log23
loga(xr) = r*logax
If the argument x of the log has an exponent r, the exponent can be moved to the front of the logarithm.
loga(1/x) = -logax
Think about the argument. (1/x) is equal to x-1. Basically this is another version of the previous property.
log2(1/3) = -log23
logaa = 1
If the base a equals the argument a the answer is 1. This is very easy to remember if one thinks about the logarithm in exponential form. How many times should one multiply a by itself to get a? Once.
log22 = 1
loga1 = 0
If the argument is one the answer is always zero. This property holds true because any number with an exponent of zero is equal to one.
(logbx/logba) = logax
This is known as "Change of Base". One log divided by another, both with the same base b, is equal to a single log. The argument a of the denominator becomes the new base, and the argument x of the numerator becomes the new argument. This is easy to remember if you think about the base as the bottom of an object and the denominator as the bottom of a fraction.
log25 = (log 5/log 2)
Tips and Warnings:
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