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Let's work through the questions in the image you uploaded.

### 31–35: Express in Decimal and Exponential Notation

Here are the decimal and exponential representations:

31. \( \frac{1}{10} = 0.1 = 1.0 \times 10^{-1} \)  
32. \( \frac{1}{100} = 0.01 = 1.0 \times 10^{-2} \)  
33. \( \frac{1}{1000} = 0.001 = 1.0 \times 10^{-3} \)  
34. \( \frac{1}{1,000,000} = 0.000001 = 1.0 \times 10^{-6} \)  
35. \( \frac{1}{10,000} = 0.0001 = 1.0 \times 10^{-4} \)  

### 36–40: Logarithms of Numbers

Here are the logarithmic values:

36. \( 100 = 10^2 \), so \( \log_{10} 100 = 2 \)  
37. \( 10 = 10^1 \), so \( \log_{10} 10 = 1 \)  
38. \( 1 = 10^0 \), so \( \log_{10} 1 = 0 \)  
39. \( 0.1 = 10^{-1} \), so \( \log_{10} 0.1 = -1 \)  
40. \( 0.01 = 10^{-2} \), so \( \log_{10} 0.01 = -2 \)

Would you like a detailed explanation of any of these? Let me know if you have any further questions.

Here are 5 related questions for further practice:

1. What is \( \frac{1}{1,000,000,000} \) in decimal and exponential form?
2. What is the logarithm of \( 1000 \)?
3. How do you express \( \frac{1}{100,000} \) in both decimal and exponential notation?
4. What is the value of \( \log_{10} 0.001 \)?
5. What does \( \log_{10} 10,000 \) equal?

**Tip**: For logarithms, remember that \( \log_{10} \) (log base 10) of a number tells you the exponent to which 10 must be raised to get that number.

Express the following in both decimal and exponential notation (31-35). Give the logarithms of the following numbers (36-40).

Learn how to express numbers in decimal and exponential notation, and calculate the logarithms of base-10 numbers. This problem covers key mathematical concepts such as exponential notation and logarithmic identities. Suitable for students in Grades 8-10.

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