Math Problem Statement

If log 27 = 1.431, then the value of log 9 is:

a. 0.934

b. 0.945

c. 0.958

d. 0.954

Solution

We are given that:

log27=1.431\log 27 = 1.431

Our task is to find the value of log9\log 9.

First, express both 27 and 9 in terms of powers of 3: 27=33and9=3227 = 3^3 \quad \text{and} \quad 9 = 3^2

Using the logarithmic identity: log(ab)=bloga\log(a^b) = b \cdot \log a

We can express log27\log 27 and log9\log 9 as follows: log27=log(33)=3log3\log 27 = \log(3^3) = 3 \cdot \log 3 log9=log(32)=2log3\log 9 = \log(3^2) = 2 \cdot \log 3

From log27=1.431\log 27 = 1.431, we know: 3log3=1.4313 \cdot \log 3 = 1.431

Solving for log3\log 3: log3=1.4313=0.477\log 3 = \frac{1.431}{3} = 0.477

Now, substitute log3\log 3 into the expression for log9\log 9: log9=2log3=20.477=0.954\log 9 = 2 \cdot \log 3 = 2 \cdot 0.477 = 0.954

Thus, the value of log9\log 9 is:

0.954\boxed{0.954}

Answer: d. 0.954

Do you have any questions or want more details on the steps?

Here are 5 questions to further explore logarithms:

  1. What is the logarithmic property that relates the logarithm of a product to individual logarithms?
  2. How would you find log81\log 81 if you know log27\log 27?
  3. What is the change of base formula for logarithms, and how is it used?
  4. How does the value of log9\log 9 change if the base is 2 instead of 10?
  5. Can you explain why logarithms are useful in solving exponential equations?

Tip: When working with logarithms, always try to express the numbers in terms of powers of the same base for simpler calculations.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Algebra
Powers and Roots

Formulas

log(a^b) = b * log a

Theorems

Logarithmic Identity for Exponents

Suitable Grade Level

Grades 10-12