Properties of Logarithms
You should be familiar with the Laws of Exponents,
and with the definition of the logarithm function.
In this lesson, we will find Laws of Logarithms
that correspond to Laws of Exponents.
Recall the definition of the logarithmic function
with base b. If b^m=x, then m is the exponent
you put on b to get x.
What is log(b) of 1? For any base b, b^0=1,
therefore 0 is the exponent you put on b to
get 1.
What is log(b) of b? For any base b, b^1=1,
therefore 1 is the exponent you put on b to
get b.
Before we go further, it’s time for a riddle:
What is the color of the blue balloon? It’s
not a difficult riddle, the description of
the object gives the information necessary
to answer the question. The color of the blue
balloon is the color blue.
There are two rules which combine exponents
and logarithms. What is log(b)b^m? We can
use our definitions to solve this, b^m=x and
log(b)x=m, but already knew this from the
description. The exponent you put on b to
get b with an exponent of m is m.
This one is similar. b raised the the power
log(b)x can be found using our definitions.
We can also get the answer directly, b is
raised to the power, which is the power to
which you raise b to get x.
What is log(b) of 1/x? Using the definition,
x = b^m so 1/x=1/b^m and we know that reciprocals
are represented by negative exponents. Therefore,
the exponent you put on b to get 1/x is −m,
which is the opposite of the exponent you
put on b to get x.
We will now define a second exponential and
logarithmic equation and ask what happens
when we multiply. What exponent do we put
on b to get xy? When multiplying numbers,
we add exponents. So m + n is the exponent
we put on b to get xy.
What exponent do we put on b to get x/y? When
dividing numbers, we subtract exponents. So
m − n is the exponent we put on b to get
x/y.
Finally, how do we handle exponents within
logarithms. Using our definitions, x^n=(b^m)^n=b^mn=b^(n*m),
so n times m is the exponent we put on b to
get x^n.
To recap: Recall that logs are exponents.
Zero is the exponent you put on anything to
get 1. An exponent of 1 leaves the base the
same. Exponents and logarithms are inverses,
so they undo each other. Negative logs are
negative exponents, which correspond to reciprocals.Multiplying
numbers corresponds to adding exponents, and
therefore to adding logs. Dividing numbers
corresponds to subtracting exponents, and
therefore to subtracting logs. When raising
a power to a power, the exponents are multiplied.
