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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