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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 4 - 2005 - Paper 2

Question 4

Find, giving your answer to 3 significant figures where appropriate, the value of x for which (a) $3^x = 5$. (b) $\log_2(2x + 1) - \log_2(x) = 2$.

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 4 - 2005 - Paper 2

Step 1

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Answer

To solve for $x$ , we can take the logarithm base 3 on both sides:

$\log_3(3^x) = \log_3(5)$

This simplifies to:

$x = \log_3(5)$

Using the change of base formula:

$x = \frac{\log_{10}(5)}{\log_{10}(3)}$

Calculating this using a calculator, we find:

$x \approx 1.464$

Thus, to three significant figures, the value of $x$ is:

$x \approx 1.46$ .

Step 2

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104 rated

Answer

Using the properties of logarithms, we can combine the logarithms:

$\log_2\left(\frac{2x + 1}{x}\right) = 2$

Exponentiating both sides gives:

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

Now, multiply both sides by $x$ to eliminate the fraction:

$2x + 1 = 4x$

Rearranging yields:

$1 = 4x - 2x$

So:

$2x = 1\quad \Rightarrow \quad x = \frac{1}{2} = 0.5$

Therefore, the value of $x$ is:

$x = 0.5$ .

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