Evaluate logarithmic and exponential expressions on this math worksheet.

Let's evaluate each logarithmic expression step by step.

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1. $ \log_7 49 $

We want to find the exponent to which 7 must be raised to get 49.

$$
7^x = 49 \Rightarrow 7^x = 7^2 \Rightarrow x = 2
$$

✔ Answer: $ \boxed{2} $

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2. $ \log_2 16 $

Find $ x $ such that $ 2^x = 16 $

$$
2^x = 16 = 2^4 \Rightarrow x = 4
$$

✔ Answer: $ \boxed{4} $

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3. $ \log_7 \frac{1}{49} $

Note: $ \frac{1}{49} = 7^{-2} $

So,
$$
\log_7 (7^{-2}) = -2
$$

✔ Answer: $ \boxed{-2} $

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4. $ \log_4 \frac{1}{256} $

First, express 256 as a power of 4:

$$
4^x = 256 \Rightarrow \text{But } 4 = 2^2, \text{ so } 4^x = (2^2)^x = 2^{2x}
$$
$$
256 = 2^8 \Rightarrow 2^{2x} = 2^8 \Rightarrow 2x = 8 \Rightarrow x = 4
$$

So $ 4^4 = 256 $. Therefore,
$$
\log_4 \left( \frac{1}{256} \right) = \log_4 (4^{-4}) = -4
$$

✔ Answer: $ \boxed{-4} $

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5. $ \log_9 3 $

We want $ x $ such that $ 9^x = 3 $

But $ 9 = 3^2 $, so:
$$
(3^2)^x = 3 \Rightarrow 3^{2x} = 3^1 \Rightarrow 2x = 1 \Rightarrow x = \frac{1}{2}
$$

✔ Answer: $ \boxed{\frac{1}{2}} $

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6. $ \log_5 125 $

$ 125 = 5^3 $, so:
$$
\log_5 (5^3) = 3
$$

✔ Answer: $ \boxed{3} $

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7. $ \log 1000 $

This is common logarithm (base 10):
$$
\log_{10} 1000 = \log_{10} (10^3) = 3
$$

✔ Answer: $ \boxed{3} $

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8. $ \log_2 32 $

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

✔ Answer: $ \boxed{5} $

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9. $ \log \frac{1}{100} $

Again, base 10:
$$
\log_{10} \left( \frac{1}{100} \right) = \log_{10} (10^{-2}) = -2
$$

✔ Answer: $ \boxed{-2} $

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10. $ \log_{16} 4 $

We want $ x $ such that $ 16^x = 4 $

But $ 16 = 4^2 $, so:
$$
(4^2)^x = 4 \Rightarrow 4^{2x} = 4^1 \Rightarrow 2x = 1 \Rightarrow x = \frac{1}{2}
$$

Alternatively, write in powers of 2:
- $ 16 = 2^4 $
- $ 4 = 2^2 $

So:
$$
\log_{2^4} (2^2) = \frac{2}{4} = \frac{1}{2}
$$

✔ Answer: $ \boxed{\frac{1}{2}} $

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✔ Final Answers:

| Expression | Answer |
|----------|--------|
| $ \log_7 49 $ | $ \boxed{2} $ |
| $ \log_2 16 $ | $ \boxed{4} $ |
| $ \log_7 \frac{1}{49} $ | $ \boxed{-2} $ |
| $ \log_4 \frac{1}{256} $ | $ \boxed{-4} $ |
| $ \log_9 3 $ | $ \boxed{\frac{1}{2}} $ |
| $ \log_5 125 $ | $ \boxed{3} $ |
| $ \log 1000 $ | $ \boxed{3} $ |
| $ \log_2 32 $ | $ \boxed{5} $ |
| $ \log \frac{1}{100} $ | $ \boxed{-2} $ |
| $ \log_{16} 4 $ | $ \boxed{\frac{1}{2}} $ |

Let me know if you'd like explanations using change-of-base or other methods!