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Evaluate $ rac{1}{3} ext{log}_{12} 27$. - Scottish Highers Maths - Question 6 - 2015

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Question 6

Evaluate-$-rac{1}{3}--ext{log}_{12}-27$.-Scottish Highers Maths-Question 6-2015.png

Evaluate $ rac{1}{3} ext{log}_{12} 27$.

Worked Solution & Example Answer:Evaluate $ rac{1}{3} ext{log}_{12} 27$. - Scottish Highers Maths - Question 6 - 2015

Step 1

Convert to a single logarithm

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Answer

Using the property of logarithms that states ( a \log_b c = \log_b c^a ), we can write:

13log1227=log122713=log123.\frac{1}{3} \text{log}_{12} 27 = \text{log}_{12} 27^{\frac{1}{3}} = \text{log}_{12} 3.

Step 2

Use the change of base formula

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Answer

To evaluate ( \text{log}_{12} 3 ), we convert it to a more manageable base using the change of base formula:

log123=log103log1012.\text{log}_{12} 3 = \frac{\text{log}_{10} 3}{\text{log}_{10} 12}.

Step 3

Calculate the values

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Answer

Using a calculator, we find:

  • ( \text{log}_{10} 3 \approx 0.4771 )
  • ( \text{log}_{10} 12 \approx 1.0792 )

Thus,

log123≈0.47711.0792≈0.442.\text{log}_{12} 3 \approx \frac{0.4771}{1.0792} \approx 0.442.

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