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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) $log_2(2x + 1) - log_2(x) = 2$.

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 5 - 2005 - Paper 2

Step 1

(a) $3^x = 5$

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Answer

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

$log(3^x) = log(5)$
Using the logarithmic property, this simplifies to:
$x imes log(3) = log(5)$
Thus, we find x as follows:
$x = \frac{log(5)}{log(3)}$
Calculating this using a calculator provides:
$x \approx 1.464$
Rounded to 3 significant figures, the answer is:
$x = 1.46$

Step 2

(b) $log_2(2x + 1) - log_2(x) = 2$

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Answer

We utilize the property of logarithms that states $log_a(b) - log_a(c) = log_a(\frac{b}{c})$ :

$log_2 \left( \frac{2x + 1}{x} \right) = 2$
Exponentiating both sides results in:
$\frac{2x + 1}{x} = 2^2$
This simplifies to:
$\frac{2x + 1}{x} = 4$
Multiplying both sides by x gives:
$2x + 1 = 4x$
Rearranging yields:
$1 = 4x - 2x$
$1 = 2x$
Thus,
$x = \frac{1}{2}$
Alternatively, it can also be solved as:
$x = 0.5$

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