use a LUILDI 17 log 10000 9 log2 64 10 log2 1024 11 log2 128 18 log 1000 19 log V1000 12 log2 256 20 log 10000 13 log42 21 log

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JUJ Wu Luerlises. use a LUILDI 17 log 10000 9 log2 64 10 log2 1024 11 log2 128 18 log 1000 19 log V1000 12 log2 256 20 log 10000 13 log42 21 log, 83.1 14 log: 2 15 log48 16 log: 128 22 log, 26.3 23 log1632 24 log27 81

The Correct Answer and Explanation is:

Let’s first interpret your list of logarithmic expressions and then solve each one. Afterward, I’ll provide a detailed explanation of logarithmic principles to help clarify how each solution was reached.

Correct Answers:

  1. log 10000 → 4
  2. log₂ 64 → 6
  3. log₂ 1024 → 10
  4. log₂ 128 → 7
  5. log 1000 → 3
  6. log √1000 → 1.5
  7. log₂ 256 → 8
  8. log 10000 → 4
  9. log₄ 2 → 0.5
  10. log₂ 2 → 1
  11. log₄ 8 → 1.5
  12. log₄ 128 → 3.5 (approx.)
  13. log₁₀ 83.1 → ≈ 1.919
  14. log₁₀ 26.3 → ≈ 1.420
  15. log₁₀ 1632 → ≈ 3.213
  16. log₂₇ 81 → 4/3 or ≈ 1.333

Logarithms are the inverse operations of exponentiation. In simpler terms, a logarithm answers the question: “To what exponent must a given base be raised to produce a certain number?”

The notation log_b(x) means “the power you raise b to get x.”

For example:

  • log₁₀ 1000 = 3 because $10^3 = 1000$
  • log₂ 64 = 6 because $2^6 = 64$

Here’s how we evaluate these step-by-step:

  1. log 10000: Base 10 is implied. $10^4 = 10000$, so answer is 4.
  2. log₂ 64: $2^6 = 64$, so the logarithm is 6.
  3. log₂ 1024: $2^{10} = 1024$, so it’s 10.
  4. log₂ 128: $2^7 = 128$, so it’s 7.
  5. log 1000: $10^3 = 1000$, so it’s 3.
  6. log √1000: $\sqrt{1000} = 10^{1.5}$, so log is 1.5.
  7. log₂ 256: $2^8 = 256$, so it’s 8.
  8. log₄ 2: What power of 4 gives 2? Since $4^{0.5} = 2$, the answer is 0.5.
  9. log₂ 2: $2^1 = 2$, so it’s 1.
  10. log₄ 8: Rewrite 8 as $2^3$ and 4 as $2^2$, so $log₄ 8 = log₂ 8 / log₂ 4 = 3 / 2 = 1.5$
  11. log₄ 128: Use change of base: $\log₄ 128 = \log_{10}128 / \log_{10}4 ≈ 2.107 / 0.602 ≈ 3.5$
  12. log₁₀ 83.1: Use calculator: ≈ 1.919
  13. log₁₀ 26.3: Use calculator: ≈ 1.420
  14. log₁₀ 1632: Use calculator: ≈ 3.213
  15. log₂₇ 81: Since $27 = 3^3$ and $81 = 3^4$, then: $$
    \log_{27}81 = \frac{\log_3 81}{\log_3 27} = \frac{4}{3} ≈ 1.333
    $$

Logarithms are essential in many fields, including science, engineering, and data analysis, because they simplify complex multiplications and exponential growth. They also form the basis of scales like the Richter and decibel scales.

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