PRECALCULUS WORKSHEET FOR 1 MARCH 2007

(1) Express   in logarithmic form.
● Solution :
 

2) Express 7²= 49 in logarithmic form.
● Solution: log₇49 = 2.

3) Express in exp onential form .
● Solution:

(4) Express log₅ 125 = 3 in exponential form.
● Solution: 3⁵ = 125.

(5) Find the value of
● Solution: Since

(6) Find the value of ln e².
● Solution: Since the natural logarithmic function and the
natural exponential function are inverse functions, we have
ln(e²) = 2. (Note that you can also think of doing this
via first using the power property and then the fact that
ln e = 1: ln(e²) = 2 ln e = 2 ∙ 1 = 2.)

(7) Find the value of log₄ 64.
● Solution: Since 4³ = 64, we have log₄ 64 = 3.

(8) Expand log(6x³).
● log(6x³) = log 6 + log(x³) = log 6 + 3 log x

(9) Expand
● This problem throws some interesting issues at us. We
should start by rewriting with fractional exponents instead
of radicals . This gets us to working with

Now since we have different exponents involved, we need to
first recognize that we have a product and use the product
property to get into the form

Now we can use the power property on each of them and
have

We actually have another power property to do on the
first term to get the " squared " off of the x, and we have a
product property in the second term. This leads to

We can fully expand this out to

(10) Expand ln

● Here we want to start with the quotient property, since all
of the exponents are different. This gets us to

Now we can use the power property on the first term and
have

Now we have to use the product property on the second
term since the powers are still different, so we have

Next, we're in a position to use the power property on the
remaining terms:

Finally, we distribute the minus sign through and have

(11) Condense
● The first step we want to take when dealing with condens-
ing is to use the power property to ensure that all of our log-
arithms are expressed with coefficients of +1 or -1. This
gives us

Now we can start combining. Let's start with the first
two terms . This is a sum of logarithms , which the product
property says can be written as the logarithm of a product.
We get

Now we have a difference of logarithms, which means we
want to use the quotient property and get

which is as condensed as we can get. (There's no need to
multiply out the numerator of the rational function inside
the logarithm.)

(12) Condense

● Again, our first step is to use the power property to move
up coefficients. This gets us to

Now we start condensing. Let's condense the first two
terms into a single term via the product property.

Now we can combine the last two terms using the quotient
property and have

Finally, we again use the product property to condense the
sum and have

(13) Condense

● We again start by using the power property to move up
the exponents. This gives

Now we use the quotient property on the first two terms
to get to

We now proceed by using the product property to further
condense and have

(14) Condense

● Here we want to begin inside the brackets, where we al-
ready have coefficients of 1 on the logarithms, so we can
condense via the product property to have

Now we need to use the product property to move up the
coefficients and have

Finally, we use the quotient property and have

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