Laws of logarithms (§§5.5&5.6)

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 correspond to any of the rules of exponents that you should have learnt before, 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). There is an exponential version of this rule, but it still involves logarithms:

b^x = 10^x log b.

Again, you can can use any base you like for the logarithm, but then you need to replace the 10 with that base as well. (Changing everything to base e is particularly useful when doing Calculus.)

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 (after factoring as much as possible). If you have different bases in the same problem, then pick one and use a change of base formula to convert the others. That said, if the variable only appears in an equation once, then you can still always solve it by reversing all of the operations applied to the variable, which is often simpler.

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