3.4. Properties of Logs http://www.ck12.org
3.4 Properties of Logs
Here you will be introduced to logarithmic expressions and will learn how they can be combined using properties of
arithmetic.
Log functions are inverses of exponential functions. This means the domain of one is the range of the other. This is
extremely helpful when solving an equation and the unknown is in an exponent. Before solving equations, you must
be able to simplify expressions containing logs. The rules of exponents are applied, but in non-obvious ways. In
order to get a conceptual handle on the properties of logs, it may be helpful to continually ask, what does a log
expression represent? For example, what does log 101 ,000 represent?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/60998http://www.youtube.com/watch?v=SxF44olWTyk James Sousa: Properties of Logarithms
Guidance
Exponential and logarithmic expressions have the same 3 components. They are each written in a different way so
that a different variable is isolated. The following two equations are equivalent to one another.
bx=a↔logba=x
The exponential equation is read “bto the powerxisa.” The logarithmic equation is read “log basebofaisx”.
The two most common bases for logs are 10 ande. At the PreCalculus levellogby itself implies log base 10 and
lnimplies basee.lnis called the natural log. One important restriction for all log functions is that they must have
strictly positive numbers in their arguments. So, if you press log -2 or log 0 on your calculator, it will give an error.
There are four basic properties of logs that correlate to properties of exponents.
Addition/Multiplication:
logbx+logby=logb(x·y)
bw+z=bw·bz
Subtraction/Division:
logbx−logby=logb
(x
y)
bw−z=bbwz
Exponentiation:
logb(xn) =n·logbx