Properties of Logarithms

1
Properties of Logarithms
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Since logs and exponentials of the same base are
inverse functions of each other they undo each
other.
Remember that
This means that
inverses undo each each other
7
5
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Properties of Logarithms

1.
2.
3.

CONDENSED
EXPANDED



(these properties are based on rules of exponents
since logs exponents)
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Using the log properties, write the expression as
a sum and/or difference of logs (expand).
When working with logs, re-write any radicals as
rational exponents.
using the second property
using the first property
using the third property
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Using the log properties, write the expression as
a single logarithm (condense).
using the third property
using the second property
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More Properties of Logarithms


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There is an answer to this and it must be more
than 3 but less than 4, but we can't do this one
in our head.
(2 to the what is 8?)
Let's put it equal to x and we'll solve for x.
Change to exponential form.
(2 to the what is 16?)
use log property take log of both sides (we'll
use common log)
(2 to the what is 10?)
use 3rd log property
Check by putting 23.32 in your calculator (we
rounded so it won't be exact)
solve for x by dividing by log 2
use calculator to approximate
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Example for TI-83
If we generalize the process we just did we come
up with the
Change-of-Base Formula
The base you change to can be any base so
generally well want to change to a base so we
can use our calculator. That would be either
base 10 or base e.
common log base 10
natural log base e
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Use the Change-of-Base Formula and a calculator
to approximate the logarithm. Round your answer
to three decimal places.
Since 32 9 and 33 27, our answer of what
exponent to put on 3 to get it to equal 16 will
be something between 2 and 3.
put in calculator