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Find, giving your answer to 3 significant figures where appropriate, the value of x for which (a) $3^x = 5$ - Edexcel - A-Level Maths Pure - Question 5 - 2005 - Paper 2

Question 5

Find, giving your answer to 3 significant figures where appropriate, the value of x for which (a) $3^x = 5$. (b) $ ext{log}_2(2x + 1) - ext{log}_2 x = 2$.

Worked Solution & Example Answer:Find, giving your answer to 3 significant figures where appropriate, the value of x for which (a) $3^x = 5$ - Edexcel - A-Level Maths Pure - Question 5 - 2005 - Paper 2

Step 1

(a) $3^x = 5$

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Answer

To solve the equation $3^x = 5$ , we can take logarithms of both sides:

$ext{log}_3(3^x) = ext{log}_3(5)$

This simplifies to:

$x = ext{log}_3(5)$

Using the change of base formula, we get:

$x = \frac{ ext{log}_{10}(5)}{ ext{log}_{10}(3)}$

Calculating this yields:

$x \approx 1.464$

Thus, to 3 significant figures, $x = 1.46$ .

Step 2

(b) $ ext{log}_2(2x + 1) - ext{log}_2 x = 2$

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Answer

We can use the properties of logarithms to combine the logs:

$ext{log}_2 \left( \frac{2x + 1}{x} \right) = 2$

By converting the logarithmic equation to its exponential form, we have:

$\frac{2x + 1}{x} = 2^2$

which simplifies to:

$\frac{2x + 1}{x} = 4$

Multiplying both sides by $x$ gives:

$2x + 1 = 4x$

Rearranging yields:

$1 = 4x - 2x$

which results in:

$1 = 2x$

Thus,

$x = \frac{1}{2}$

To check if there are other solutions:

$x = 0.5$ .

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