How to Evaluate Logarithms

How to Evaluate Logarithms

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If we know the squares, cubes, and roots of numbers, we can figure out the answers to many logarithms mentally. For example, consider \(log_4 \ 64\). We ask, "How many times do you have to multiply \(4\) to get \(64\)?" Since we already know \(4^3 \ = \ 64\), it stands to reason that \(log_4 \ 64 \ = \ 3\).

Even logarithms that seem more complicated can be worked out without a calculator. For example, let's evaluate \(log_{\frac{5}{6}} \ \frac{125}{216}\) mentally.

We ask, "How many times do you have to multiply \(\frac{5}{6} \ \) to get \(\frac{125}{216} \ \)?"
We know that: \(5^3 \ = \ 125\), \(6^3 \ = \ 216\). So, \(\frac{125}{216} \ = \ (\frac{5}{6} \ )^3\) \(⇒ \ log_{\frac{5}{6}} \ \frac{125}{216} \ = \ 3\)

Steps to Evaluate Logarithms Mentally

Consider: \(log_b \ x \ = \ y\):

• Rewrite the exponential expression "\(x\)" as a power of "\(b\)": \(b^y \ = \ x\)
• Use what you know about powers to figure out what \(y\) is by asking, "To what exponent should \(y\) be raised to get \(x\)?"

Learn a few rules about logarithms:

• \(log_b \ (x) \ = \ \frac{log_d \ (x)}{log_d \ (b)}\)
• \(log_a \ (x^b) \ = \ b \ log_a \ x\)
• \(log_a \ 1 \ = \ 0\)
• \(log_a \ a \ = \ 1\)

Example

Evaluate: \(log_3 \ 243\)

Solution:

Write \(243\) in the form of a power of the base, \(243 \ = \ 3^5\), then: \(log_3 \ 243 \ = \ log_3 \ 3^5\)
Use the log rule: \(log_a \ (x^b) \ = \ b \ log_a \ x\) \(⇒ \ log_3 \ 3^5 \ = \ 5log_3 \ 3\)
Use the log rule: \(log_a \ a \ = \ 1\) \(⇒ \ 5log_3 \ 3 \ = \ 5 \times 1 \ = \ 5\)

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