Let's solve each part of the problem step by step, determining the logarithm values between two integers and calculating their approximate values to four decimal places.

Part 1: (a) $\log 628,459$

• The logarithm of $628,459$ is between $10^5 = 100,000$ and $10^6 = 1,000,000$, so: $5 < \log 628,459 < 6$
• Calculate the value: $\log 628,459 \approx \log_{10}(628459) = 5.7978$

Part 2: (b) $\log 1,657,892$

• The logarithm of $1,657,892$ is between $10^6 = 1,000,000$ and $10^7 = 10,000,000$, so: $6 < \log 1,657,892 < 7$
• Calculate the value: $\log 1,657,892 \approx \log_{10}(1657892) = 6.2187$

Part 3: (c) $\log 0.22$

• The logarithm of $0.22$ is between $10^{-1} = 0.1$ and $10^0 = 1$, so: $-1 < \log 0.22 < 0$
• Calculate the value: $\log 0.22 \approx \log_{10}(0.22) = -0.6576$

Part 4: (d) $\log 0.000008$

• The logarithm of $0.000008$ is between $10^{-6} = 0.000001$ and $10^{-5} = 0.00001$, so: $-6 < \log 0.000008 < -5$
• Calculate the value: $\log 0.000008 \approx \log_{10}(0.000008) = -5.0969$

Part 5: (e) $\log (6.2 \times 10^3)$

• Split into components: $\log(6.2 \times 10^3) = \log(6.2) + \log(10^3) = \log(6.2) + 3$
• Approximate $\log(6.2)$: $\log(6.2) \approx 0.7924$
• Combine: $\log(6.2 \times 10^3) = 0.7924 + 3 = 3.7924$
• The logarithm lies between: $3 < \log (6.2 \times 10^3) < 4$

Part 6: (f) $\log (0.9 \times 10^{-5})$

• Split into components: $\log(0.9 \times 10^{-5}) = \log(0.9) + \log(10^{-5}) = \log(0.9) - 5$
• Approximate $\log(0.9)$: $\log(0.9) \approx -0.0458$
• Combine: $\log(0.9 \times 10^{-5}) = -0.0458 - 5 = -5.0458$
• The logarithm lies between: $-6 < \log (0.9 \times 10^{-5}) < -5$

Final Results:

1. (a) $5 < \log 628,459 < 6$, $\log 628,459 \approx 5.7978$
2. (b) $6 < \log 1,657,892 < 7$, $\log 1,657,892 \approx 6.2187$
3. (c) $-1 < \log 0.22 < 0$, $\log 0.22 \approx -0.6576$
4. (d) $-6 < \log 0.000008 < -5$, $\log 0.000008 \approx -5.0969$
5. (e) $3 < \log (6.2 \times 10^3) < 4$, $\log(6.2 \times 10^3) \approx 3.7924$
6. (f) $-6 < \log (0.9 \times 10^{-5}) < -5$, $\log(0.9 \times 10^{-5}) \approx -5.0458$

Would you like more explanation or any of these steps clarified?

5 Related Questions:

1. What is the relationship between logarithms and exponents?
2. How do you determine the range of a logarithm before calculating it?
3. Why does $\log(0.22)$ result in a negative number?
4. How does $\log(a \times 10^b)$ simplify to $\log(a) + b$?
5. How does the logarithm function behave for very large and very small numbers?

Tip:

Always break down logarithmic problems into smaller, simpler components, such as powers of 10, to make estimates easier before calculating precise values.