Functional Analysis: Section 4.4
Expanding and Condensing Logs
Name: ________________________
Date:____________ Period: ______
1.) Please evaluate mentally the following:
a.) log39
b.) log381
c.) Let’s consider a toughie…. log3729
d.) Boy oh boy that last one was mean! Let’s try to break 729 down into the product of
smaller values. HINT: Consider your work in a and b.
log3729 = log3
e.) How else could we express the right side of your equation? Is there a way to simplify the
right side of the equation? We aren’t certain here…we are trying to determine equivalence.
log3729 =
f.) A good question is how are the two ghostie expressions above related? In order to find out
how, let’s evaluate the ghosties individually.
log3729 =
g.) What operation should we perform on the evaluations of the ghostie expressions? Should
we find the product? The difference? The ratio? HINT: Feel free to use your calculators to
help by evaluating the left side of the equation and comparing results.
2.) a.) Let’s try that again by breaking the 64 in the expression log264 into a product of
smaller values.
log264 = log2
b.) Now break the right side into two log expressions.
log264 =
c.) Evaluate the individual log expressions.
log264 =
d.) Which is it? To add, subtract, multiply, divide?
3.) Attempt to generalize the work from example 1 and 2:
log(A∙B) =
4.) Writing a logarithm as the operation of two logs is known as expanding logarithms. Try
to expand the following logarithmic expressions given your generalization in number 3.
a.) log345
b.) log432
c.) log53x
d.) log200
Rewriting multiple logarithms with the same base as a single logarithm is known as
condensing logarithms. Try to condense each of the following logarithms as a single log.
e.) log525 + log 5 15 + log51
f.) log34 + log3x
g.) log10 + log1000
5.) Consider the expression given in 2a.). Find another means to express 64 but not as the
product of two numbers. Hint: exponents…
log264 = log2
How could the log expression to the right of the equation lead me directly to the evaluation
of log264?
6.) Interesting…lets investigate further with the following. Try to rewrite each log expression
so that the exponent expression would lead directly to the evaluation of the logarithm
a.) log381 (write 81 as a square )
b.) log5625 (625 as a square) c.) log2256
d.) log2512 (512 as a cube)
e.) log264 (this time as a cube)
7.) Attempt to generalize the work from example 5 and 6:
log(AB) =
8.) Expand each of the following
a.) log243
b.) logx5
c.) log464x2
d.) log394
9.) Condense each of the following
a.) 10logx
b.) log32 + ½ log39
c.) 7log42 + 5log4 x + 3log4 y