Logarithm practice worksheet with problems and blank answer spaces.

Let's solve each logarithmic expression step by step using logarithmic properties and basic exponent rules.

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Logarithm Basics:

Recall:
- $\log_b(b) = 1$
- $\log_b(1) = 0$ for any base $b > 0$, $b \ne 1$
- $\log_b(a^n) = n \cdot \log_b(a)$
- $\log_b\left(\frac{1}{a}\right) = -\log_b(a)$
- If $b^x = a$, then $\log_b(a) = x$

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Now, let’s evaluate each one:

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1. $\log_7(49)$
$49 = 7^2$, so $\log_7(49) = \log_7(7^2) = 2$

✔ Answer: 2

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2. $\log_2(32)$
$32 = 2^5$, so $\log_2(32) = \log_2(2^5) = 5$

✔ Answer: 5

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3. $\log_{12}(1)$
Any log of 1 is 0, since $b^0 = 1$

✔ Answer: 0

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4. $\log_4\left(\frac{1}{16}\right)$
$\frac{1}{16} = \frac{1}{4^2} = 4^{-2}$, so $\log_4(4^{-2}) = -2$

✔ Answer: -2

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5. $\log_3(3)$
$\log_b(b) = 1$, so this is 1

✔ Answer: 1

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6. $\log_4(16)$
$16 = 4^2$, so $\log_4(16) = \log_4(4^2) = 2$

✔ Answer: 2

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7. $\log_2\left(\frac{1}{2}\right)$
$\frac{1}{2} = 2^{-1}$, so $\log_2(2^{-1}) = -1$

✔ Answer: -1

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8. $\log_{10}(100)$
$100 = 10^2$, so $\log_{10}(100) = 2$

✔ Answer: 2

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9. $\log_4(4)$
$\log_b(b) = 1$, so this is 1

✔ Answer: 1

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10. $\log_2(2)$
$\log_b(b) = 1$, so this is 1

✔ Answer: 1

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11. $\log_4(1)$
$\log_b(1) = 0$, so this is 0

✔ Answer: 0

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12. $\log_3(81)$
$81 = 3^4$, so $\log_3(81) = \log_3(3^4) = 4$

✔ Answer: 4

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13. $\log_{10}\left(\frac{1}{1000}\right)$
$\frac{1}{1000} = 10^{-3}$, so $\log_{10}(10^{-3}) = -3$

✔ Answer: -3

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14. $\log_7(7)$
$\log_b(b) = 1$, so this is 1

✔ Answer: 1

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15. $\log_3(9)$
$9 = 3^2$, so $\log_3(9) = \log_3(3^2) = 2$

✔ Answer: 2

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16. $\log_7(1)$
$\log_b(1) = 0$, so this is 0

✔ Answer: 0

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17. $\log_{10}\left(\frac{1}{100}\right)$
$\frac{1}{100} = 10^{-2}$, so $\log_{10}(10^{-2}) = -2$

✔ Answer: -2

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18. $\log_7\left(\frac{1}{7}\right)$
$\frac{1}{7} = 7^{-1}$, so $\log_7(7^{-1}) = -1$

✔ Answer: -1

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19. $\log_8(1)$
$\log_b(1) = 0$, so this is 0

✔ Answer: 0

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20. $\log_7\left(\frac{1}{49}\right)$
$\frac{1}{49} = \frac{1}{7^2} = 7^{-2}$, so $\log_7(7^{-2}) = -2$

✔ Answer: -2

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✔ Final Answers (in order):

| Expression | Answer |
|----------|--------|
| $\log_7(49)$ | 2 |
| $\log_2(32)$ | 5 |
| $\log_{12}(1)$ | 0 |
| $\log_4\left(\frac{1}{16}\right)$ | -2 |
| $\log_3(3)$ | 1 |
| $\log_4(16)$ | 2 |
| $\log_2\left(\frac{1}{2}\right)$ | -1 |
| $\log_{10}(100)$ | 2 |
| $\log_4(4)$ | 1 |
| $\log_2(2)$ | 1 |
| $\log_4(1)$ | 0 |
| $\log_3(81)$ | 4 |
| $\log_{10}\left(\frac{1}{1000}\right)$ | -3 |
| $\log_7(7)$ | 1 |
| $\log_3(9)$ | 2 |
| $\log_7(1)$ | 0 |
| $\log_{10}\left(\frac{1}{100}\right)$ | -2 |
| $\log_7\left(\frac{1}{7}\right)$ | -1 |
| $\log_8(1)$ | 0 |
| $\log_7\left(\frac{1}{49}\right)$ | -2 |

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Let me know if you'd like these explained in more detail!