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Given that log₄ 36 - log₄ a = rac{1}{2}, find the value of a. - Scottish Highers Maths - Question 12 - 2017

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Question 12

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Given that log₄ 36 - log₄ a = rac{1}{2}, find the value of a.

Worked Solution & Example Answer:Given that log₄ 36 - log₄ a = rac{1}{2}, find the value of a. - Scottish Highers Maths - Question 12 - 2017

Step 1

Use laws of logs

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Answer

According to the laws of logarithms, we can combine the two logarithmic expressions:

egin{align*} log_4 36 - log_4 a & = log_4 \left(\frac{36}{a}\right) \ \end{align*}

Thus, we rewrite the equation as:

log4(36a)=12log_4 \left(\frac{36}{a}\right) = \frac{1}{2}

Step 2

Write in exponential form

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Answer

Next, we convert the logarithmic equation into its exponential form:

36a=41/2\frac{36}{a} = 4^{1/2}

Since 41/24^{1/2} is the same as 22, the equation simplifies to:

36a=2\frac{36}{a} = 2

Step 3

Solve for a

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Answer

We can now solve for aa by rearranging the equation:

a=362a = \frac{36}{2}

Calculating this gives:

a=18a = 18

Thus, the final value of a is:

18

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