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amyyzhu · Jul 9, 2012, 11:18 PM

How do you simplify logs with fractions or roots? With normal simple numbers, I understand but using fractions or roots, I get really confused. I think it's also because I'm rocky with indices as a whole to start with so that didn't help.

Example;
Given that logb(2) = 0.3010 and logb(3) = 0.4771, find: logb(root 2) & logb(1/3)

Thank you in advance!

ebaines · Jul 10, 2012, 06:29 AM

As you study logarithms you really must memorize and become comfortable with the key relationships, which include:

$ \log _a (bc) = \log_a b + \log_a c\\ \log _a (\frac b c } = \log _a b - \log _a c\\ \log _a b^n = n \log _a b\\ \log_a b \ \times\ \log _b c = \log _a c\\ a^{\log_a b} = b $

Using the third relationship on your first example you have:

$ \log_b \sqrt 2 = \log _b 2^{1/2} = \frac 1 2 \log_b 2 $

For the log of 1/3 you can do it two different ways. First using the relationship for $\log(a^n)$ again:

$ \log_b \frac 1 3 = \log_b 3 ^{-1} = -1 \log _b 3 $

Or using the relationship for $\log (a/b)$ :
$ \\log_b \frac 1 3 = \log_b 1 - \log_b 3. $

Since $\log 1 = 0$ , this leads to the same result.