OUTCOME B13: Demonstrate An Understanding Of The Properties Of Logarithms And Apply Them

Originators:

Emily and Isaac

Explanation:

A logarithm is the inverse of an exponent, or simply put, logs are the opposites of exponents. At first it might sound complicated, but once you know the formula, it’s quite simple. Some exponentials are way too difficult to solve for using exponent rules because they are big decimals and irrational numbers, and this is where logarithms come into play. The first formula of logarithms is  y = b^x is equivalent to logb(y) = x  As you can notice here, the bases stay the same on both versions of the equation. When there is no base stated, it is always 10. When you are solving for a logarithm, you need to use a calculator. There should be two log options on each calculator. One will have a “10” on it or on a second function, and the other will have an “E”. The one we will use will be the one with the “10”. This means that the log has a base of ten, so solving without a base of ten would not work. Most equations you are solving won’t have a base of 10, but there is a formula you can use to change that called the Change of Base Formula: logab = log b

                                                                                                                    log a

There are also basic logarithm rules which can help you solve logarithmic equations, but these are only for logs with the same bases. There are three rules which can all be reversed.

1. Addition - Multiplication: logA + logB = logA x B

2. Subtraction – Division: logA – logB = logA

                                                                   B

3. Exponent – Multiplication: logA^B = BlogA

Sample Problems:

The Giant Rabid Mongoose was so displeased at not catching Joe-Bob due to his excellent jet-pack skills that he went on a rampage. He yelled, screamed, jumped, destroyed villages, ate mountains, and everything that a Giant Rabid Mongoose Rampage includes. As time went by, more and more got destroyed, and he was quickly growing angrier. With each village he destroyed, his anger increased exponentially, due to the lack of satisfaction in Joe-Bob and his jet-pack not being included in the massacre. When Joe-Bob first flew away, GRM’s anger was 1000 on the anger scale, and as he destroyed villages at a magically consistent rate, his anger had increased to 12 300 on the scale by the sixth village.

a. Write an equation for the Giant Rabid Mongoose’s anger.

b. How many villages would he have had to destroy for his anger to reach 15 000?

Solve:

c. 2log44 – log42 + log49

That Giant Rabid Mongoose has really done it this time.

Solutions:

a. 12 300                                        y = 1000(12.3)^x

      1000  = 12.3              so                                     6

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b. 15000 = 1000(12.3)^x

                                      6

     1000          1000

15 = (12.3)^x

                    6

x = log12.315

6

x = log15

6    log12.3

(6) x = 1.08 (6)

     6

x = 6.48

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c. log44^2 – log42 + log49

log416 - log42 + log49

log416 + log49

        2

log48 x 9

log472

log72

log4

3.08

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PS: The Giant Rabid Mongoose destroyed the whole world and all of humanity, except for Joe-Bob, who flew off into space on his jet-pack.