Let’s go through each section of the worksheet step by step. I’ll solve every problem carefully and check my work as I go.
---
Section 1: Rewrite the exponential equation in logarithmic form.
Remember:
If you have \( a^b = c \), then the log form is \( \log_a c = b \).
For natural logs (base e): \( e^x = y \) becomes \( \ln y = x \).
1. \( 5^3 = 125 \) → \( \log_5 125 = 3 \)
✔
2. \( 6^{-2} = \frac{1}{36} \) → \( \log_6 \frac{1}{36} = -2 \)
✔
3. \( e^3 = 20.085 \) → \( \ln 20.085 = 3 \)
✔
4. \( e^x = 4 \) → \( \ln 4 = x \)
✔
5. \( 8^2 = 64 \) → \( \log_8 64 = 2 \)
✔
6. \( 81^{1/4} = 3 \) → \( \log_{81} 3 = \frac{1}{4} \)
✔
7. \( 10^{-3} = 0.001 \) → \( \log 0.001 = -3 \) (since base 10 is implied)
✔
8. \( e^0 = 1 \) → \( \ln 1 = 0 \)
✔
9. \( u^v = w \) → \( \log_u w = v \)
✔
10. \( 9^{3/2} = 27 \) → \( \log_9 27 = \frac{3}{2} \)
✔
All correct!
---
Section 2: Rewrite the logarithmic equation in exponential form.
Remember:
If you have \( \log_a c = b \), then exponential form is \( a^b = c \).
For natural logs: \( \ln y = x \) → \( e^x = y \)
1. \( \log_2 8 = x \) → \( 2^x = 8 \)
✔
2. \( \log_5 625 = 4 \) → \( 5^4 = 625 \)
✔
3. \( \log_x 13 = 5 \) → \( x^5 = 13 \)
✔
4. \( \log_2 \frac{1}{8} = -3 \) → \( 2^{-3} = \frac{1}{8} \)
✔
5. \( \log_4 64 = 3 \) → \( 4^3 = 64 \)
✔
6. \( \ln 143 = x \) → \( e^x = 143 \)
✔
7. \( \log 1000 = 3 \) → \( 10^3 = 1000 \)
✔
8. \( \ln x = 14 \) → \( e^{14} = x \)
✔
9. \( \log \frac{1}{100} = -2 \) → \( 10^{-2} = \frac{1}{100} \)
✔
10. \( \ln 18 = x \) → \( e^x = 18 \)
✔
All correct!
---
Section 3: Use your calculator to evaluate each logarithm. Round to four decimal places.
We’ll use a calculator for these. Remember:
- “log” without a base means base 10.
- “ln” means natural log (base e).
1. \( \log 68 \) → ≈ 1.8325
✔
2. \( \log 100 \) → = 2.0000
✔
3. \( \ln 9 \) → ≈ 2.1972
✔
4. \( \log 10 \) → = 1.0000
✔
5. \( \ln 216 \) → ≈ 5.3753
✔
6. \( \ln 9548 \) → ≈ 9.1641
✔
7. \( \log 0.0001 \) → = -4.0000
✔
8. \( \log 17 \) → ≈ 1.2304
✔
9. \( \ln 125 \) → ≈ 4.8283
✔
10. \( \log 6158 \) → ≈ 3.7894
✔
All match the handwritten answers — they are all correct!
---
Final Answer:
All problems on the worksheet are solved correctly. The student has accurately rewritten equations between exponential and logarithmic forms and evaluated logarithms using a calculator with proper rounding.
Parent Tip: Review the logic above to help your child master the concept of exponential and logarithmic equations worksheet.
