## How to Evaluate Logarithms

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: $$log_a \ a \ = \ 1$$ $$⇒ \ 5log_3 \ 3 \ = \ 5 \times 1 \ = \ 5$$

### Exercises for Evaluating Logarithms

1) Find the answer: $$log_8 \ 64 \ =$$

2) Find the answer: $$log_{13} \ 169 \ =$$

3) Find the answer: $$log_{25} \ 625 \ =$$

4) Find the answer: $$log_{\frac{1}{3}} \ 729 \ =$$

5) Find the answer: $$log_{\frac{1}{5}} \ 3125 \ =$$

6) Find the answer: $$log_{6} \ 1296 \ =$$

7) Find the answer: $$log_{7} \ 343 \ =$$

8) Find the answer: $$log_4 \ 9 \ =$$

9) Find the answer: $$log_{11} \ 14641 \ =$$

10) Find the answer: $$log_{\frac{1}{17}} \ 289 \ =$$

1) Find the answer: $$log_8 \ 64 \ =$$

$$\color{red}{log_8 \ 64 \ = \ log_{2^3} \ 2^6 \ = \ \frac{6}{3} \ log_2 \ 2 \ = \ \frac{6}{3} \ = \ 2}$$

2) Find the answer: $$log_{13} \ 169 \ =$$

$$\color{red}{log_{13} \ 169 \ = \ log_{13} \ 13^2 \ = \ 2 \ log_{13} \ 13 \ = \ 2}$$

3) Find the answer: $$log_{25} \ 625 \ =$$

$$\color{red}{log_{25} \ 625 \ = \ log_{25} \ 25^2 \ = \ 2 \ log_{25} \ 25 \ = \ 2}$$

4) Find the answer: $$log_{\frac{1}{3}} \ 729 \ =$$

$$\color{red}{log_{\frac{1}{3}} \ 729 \ = \ log_{3^{-1}} \ 3^6 \ = \ \frac{6}{-1} \ log_3 \ 3 \ = \ -6}$$

5) Find the answer: $$log_{\frac{1}{5}} \ 3125 \ =$$

$$\color{red}{log_{\frac{1}{5}} \ 3125 \ = \ log_{5^{-1}} \ 5^5 \ = \ \frac{5}{-1} \ log_5 \ 5 \ = \ -5}$$

6) Find the answer: $$log_{6} \ 1296 \ =$$

$$\color{red}{log_{6} \ 1296 \ = \ log_{6} \ 6^4 \ = \ 4 \ log_6 \ 6 \ = \ 4}$$

7) Find the answer: $$log_{7} \ 343 \ =$$

$$\color{red}{log_{7} \ 343 \ = \ log_{7} \ 7^3 \ = \ 3 \ log_7 \ 7 \ = \ 3}$$

8) Find the answer: $$log_4 \ 9 \ =$$

$$\color{red}{log_4 \ 9 \ = \ \frac{log_2 \ 9}{log_2 \ 4} \ = \ \frac{log_2 \ 9}{log_2 \ 2^2} \ = \ \frac{log_2 \ 9}{ 2 \ log_2 \ 2} \ = \ \frac{log_2 \ 9}{2}}$$

9) Find the answer: $$log_{11} \ 14641 \ =$$

$$\color{red}{log_{11} \ 14641 \ = \ log_{11} \ 11^4 \ = \ 4 \ log_{11} \ 11 \ = \ 4}$$

10) Find the answer: $$log_{\frac{1}{17}} \ 289 \ =$$

$$\color{red}{log_{\frac{1}{17}} \ 289 \ = \ log_{17^{-1}} \ 17^2 \ = \ \frac{2}{-1} \ log_{17} \ 17 \ = \ -2}$$

## Evaluating Logarithms Practice Quiz

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