Properties Of Logs Worksheet - Free Printable

Here are the step-by-step solutions for the problems on the worksheet.

Part 1: Write each equation in exponential form.

Rule: To change from logarithmic form $\log_b(a) = c$ to exponential form, you write it as $b^c = a$. The base stays the base, the answer becomes the exponent.

1. Problem: $\log_2 64 = 6$
* Base is 2, exponent is 6, result is 64.
* Answer: $2^6 = 64$

2. Problem: $\log_4 \frac{1}{64} = -3$
* Base is 4, exponent is -3, result is 1/64.
* Answer: $4^{-3} = \frac{1}{64}$

3. Problem: $\log_{10} (0.01) = -2$
* Base is 10, exponent is -2, result is 0.01.
* Answer: $10^{-2} = 0.01$

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Part 2: Write each equation in logarithmic form.

Rule: To change from exponential form $b^c = a$ to logarithmic form, you write it as $\log_b(a) = c$. The base stays the base, the exponent becomes the answer.

4. Problem: $2^5 = 32$
* Base is 2, exponent is 5, result is 32.
* Answer: $\log_2 32 = 5$

5. Problem: $5^{-1/2} = \frac{\sqrt{5}}{5}$
* Base is 5, exponent is -1/2, result is $\frac{\sqrt{5}}{5}$.
* Answer: $\log_5 \left(\frac{\sqrt{5}}{5}\right) = -\frac{1}{2}$

6. Problem: $10^{-1} = 0.1$
* Base is 10, exponent is -1, result is 0.1.
* Answer: $\log_{10} (0.1) = -1$

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Part 3: Evaluate the expression.

Rule: To evaluate $\log_b(a)$, ask yourself: "Base $b$ raised to what power equals $a$?"

7. Problem: $\log_2 8$
* $2^? = 8$. Since $2 \times 2 \times 2 = 8$, the answer is 3.
* Answer: 3

8. Problem: $\log_8 64$
* $8^? = 64$. Since $8 \times 8 = 64$, the answer is 2.
* Answer: 2

9. Problem: $\log_6 216$
* $6^? = 216$. $6 \times 6 = 36$, and $36 \times 6 = 216$. The answer is 3.
* Answer: 3

10. Problem: $\log_7 7$
* $7^? = 7$. Any number to the power of 1 is itself.
* Answer: 1

11. Problem: $\log_6 1$
* $6^? = 1$. Any non-zero number to the power of 0 is 1.
* Answer: 0

12. Problem: $\log_8 \frac{1}{8}$
* $8^? = \frac{1}{8}$. A negative exponent flips the fraction. $8^{-1} = \frac{1}{8}$.
* Answer: -1

13. Problem: $\log_7 \frac{1}{49}$
* $7^? = \frac{1}{49}$. Since $7^2 = 49$, then $7^{-2} = \frac{1}{49}$.
* Answer: -2

14. Problem: $\log_9 \frac{1}{27}$
* $9^? = \frac{1}{27}$.
* Rewrite bases as powers of 3: $9 = 3^2$ and $27 = 3^3$.
* $(3^2)^x = \frac{1}{3^3} \rightarrow 3^{2x} = 3^{-3}$.
* $2x = -3 \rightarrow x = -1.5$ (or $-3/2$).
* Answer: -1.5

15. Problem: $\log_5 \sqrt{5}$
* $5^? = \sqrt{5}$. A square root is the same as the power of $1/2$.
* Answer: 0.5 (or $1/2$)

16. Problem: $\log_9 3$
* $9^? = 3$. Since $\sqrt{9} = 3$, the power is $1/2$.
* Answer: 0.5 (or $1/2$)

17. Problem: $\log_2 16$
* $2^? = 16$. $2 \times 2 \times 2 \times 2 = 16$.
* Answer: 4

18. Problem: $\log_{1/2} 16$
* $(1/2)^? = 16$.
* $(1/2)^1 = 1/2$
* $(1/2)^2 = 1/4$
* $(1/2)^3 = 1/8$
* $(1/2)^4 = 1/16$
* $(1/2)^{-4} = 2^4 = 16$.
* Answer: -4

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Part 4: Solve for x.

Rule: Convert the logarithmic equation to exponential form ($b^c = a$) and solve for the unknown variable.

19. Problem: $\log_6 x = 2$
* Convert: $6^2 = x$
* Calculate: $36 = x$
* Answer: 36

20. Problem: $\log_5 x = 3$
* Convert: $5^3 = x$
* Calculate: $125 = x$
* Answer: 125

21. Problem: $\log_{16} x = -1$
* Convert: $16^{-1} = x$
* Calculate: $x = \frac{1}{16}$
* Answer: 1/16

22. Problem: $\log_9 x = 2$
* Convert: $9^2 = x$
* Calculate: $81 = x$
* Answer: 81

23. Problem: $\log_{1/4} x = -2$
* Convert: $(1/4)^{-2} = x$
* Calculate: Flip the fraction to make the exponent positive: $4^2 = x$.
* $16 = x$
* Answer: 16

24. Problem: $\log_x 64 = 3$
* Convert: $x^3 = 64$
* Solve: What number cubed equals 64? $4 \times 4 \times 4 = 64$.
* Answer: 4

25. Problem: $\log_x 8 = -1$
* Convert: $x^{-1} = 8$
* Solve: $\frac{1}{x} = 8$. Multiply both sides by $x$ and divide by 8.
* $1 = 8x \rightarrow x = 1/8$.
* Answer: 1/8

Final Answer:
1. $2^6 = 64$
2. $4^{-3} = \frac{1}{64}$
3. $10^{-2} = 0.01$
4. $\log_2 32 = 5$
5. $\log_5 (\frac{\sqrt{5}}{5}) = -\frac{1}{2}$
6. $\log_{10} (0.1) = -1$
7. 3
8. 2
9. 3
10. 1
11. 0
12. -1
13. -2
14. -1.5
15. 0.5
16. 0.5
17. 4
18. -4
19. 36
20. 125
21. 1/16
22. 81
23. 16
24. 4
25. 1/8