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Trick Questions Based on Log Rules (page 4 of 5)

Sections: Basic log rules, Expanding, Simplifying, Trick questions, Change-of-Base formula

You should expect to need to know these log rules, because there is a certain type of question that the teacher can put on the test to make sure you know how to use the rules. Warning: You won't be able to "cheat" with your calculator.

• Let log_b(2) = 0.3869, log_b(3) = 0.6131, and log_b(5) = 0.8982. Using these values, evaluate log_b(10).

Since 10 = 2 × 5, then:

log_b(10) = log_b(2 × 5) = log_b(2) + log_b(5)

Since I have the values for log_b(2) and log_b(5), I can evaluate:

log_b(2) + log_b(5) = 0.3869 + 0.8982 = 1.2851

Then log_b(10) = 1.2851.

• Let log_b(2) = 0.3869, log_b(3) = 0.6131, and log_b(5) = 0.8982. Using these values, evaluate log_b(9).

Since 9 = 3², then:

log_b(9) = log_b(3²) = 2log_b(3)

Since I have the value for log_b(3), then I can evaluate:

2log_b(3) = 2(0.6131) = 1.2262

Then log_b(9) = 1.2262.

• Let log_b(2) = 0.3869, log_b(3) = 0.6131, and log_b(5) = 0.8982. Using these values, evaluate log_b(7.5).

This one is a bit more complicated, but, after fiddling with the numbers for a bit, I notice that 7.5 = 15 ÷ 2, so:

log_b(7.5) = log_b(15 ÷ 2) = log_b(15) – log_b(2)

And 15 = 5 × 3, so:   Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

log_b(15) – log_b(2)
     = [log_b(5) + log_b(3)] – log_b(2)
     = log_b(5) + log_b(3) – log_b(2)

And now I can evaluate:

log_b(5) + log_b(3) – log_b(2)
     = 0.8982 + 0.6131 – 0.3869
     = 1.1244

Then log_b(7.5) = 1.1244.

• Let log_b(2) = 0.3869, log_b(3) = 0.6131, and log_b(5) = 0.8982. Using these values, evaluate log_b(6).

Since 6 = 2 × 3, then:

log_b(6) = log_b(2 × 3) = log_b(2) + log_b(3)

Since I have these values, I can evaluate:

log_b(2) + log_b(3) = 0.3869 + 0.6131 = 1.0000

Then log_b(6) = 1.0000.

Hmm... that was interesting. I got that log_b(6) = 1. Using The Relationship, I get:

log_b(6) = 1
b¹ = 6
b = 6

So now I know that their mysterious unnamed base "b" was actually 6! But they will not usually give you problems that let you figure out the base like this.

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