Laws of logarithms (§§6.5&6.6)

The laws of exponents can be turned into laws of logarithms.

Law of exponents:	Law of logarithms:
b⁰ = 1,	log_b 1 = 0;
b¹ = b,	log_b b = 1;
b^x + y = b^xb^y,	log_b (uv) = log_b u + log_b v;
b^x − y = b^x/b^y,	log_b (u/v) = log_b u − log_b v;
b^xy = (b^y)^x,	log_b (u^x) = x log_b u;
b^y/x = ^x√(b^y),	log_b (^x√u) = log_b u/x.

In these rules, b can be any valid base (any positive number different from 1), x and y can be any real numbers (except that x cannot be 0 in the last rule), and u and v can be any positive numbers (which are the numbers that one can take logarithms of).

Another important rule, which doesn't directly correspond to any particular rule of exponents, is the change-of-base formula:

log_b u = (log u)/(log b).

Here, the logarithms on the right-hand side can have any base (as long as you use the same base for both), so you may as well use your favourite (which in this class is 10).

Each law of logarithms can be used in two directions: to break down the logarithm of a complicated expression into an expression involving simple logarithms, or to combine an expression into a single logarithm. When breaking down a logarithm, you may have to do some factoring.

To solve an equation involving logarithms with the same base, combine both sides into logarithms and drop the logs; to solve an equation involving variables in the exponents, take logarithms of both sides and break them down. If you have different bases in the same problem, pick one and use the change of base formula to convert the others.

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This web page was written in 2012 by Toby Bartels, last edited on 2012 November 9. Toby reserves no legal rights to it.

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