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Find, giving your answer to 3 significant figures where appropriate, the value of x for which (a) $5^x = 10$, (b) $\log_{10}(x - 2) = -1$. - Edexcel - A-Level Maths Pure - Question 5 - 2011 - Paper 2

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Find,-giving-your-answer-to-3-significant-figures-where-appropriate,-the-value-of-x-for-which--(a)-$5^x-=-10$,--(b)-$\log_{10}(x---2)-=--1$.-Edexcel-A-Level Maths Pure-Question 5-2011-Paper 2.png

Find, giving your answer to 3 significant figures where appropriate, the value of x for which (a) $5^x = 10$, (b) $\log_{10}(x - 2) = -1$.

Worked Solution & Example Answer:Find, giving your answer to 3 significant figures where appropriate, the value of x for which (a) $5^x = 10$, (b) $\log_{10}(x - 2) = -1$. - Edexcel - A-Level Maths Pure - Question 5 - 2011 - Paper 2

Step 1

(a) $5^x = 10$

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Answer

To solve for xx, we start by taking the logarithm on both sides:

ext{Taking logs:} \\ ext{log } (5^x) = ext{log } (10) \\ ext{This simplifies to:} \\ x imes ext{log } (5) = ext{log } (10) \\ x = \frac{\text{log } (10)}{\text{log } (5)} \\ \end{align*}$$ Now calculating: - Using a calculator: $$\text{log } (10) \approx 1.0000 \\ \text{log } (5) \approx 0.6990 \\ \Rightarrow x \approx \frac{1.0000}{0.6990} \approx 1.4307$$ Rounding to 3 significant figures, we get: $$x \approx 1.43$$

Step 2

(b) $\log_{10}(x - 2) = -1$

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Answer

To solve this equation, we can transform the logarithmic equation to its exponential form:

x−2=10−1x - 2 = 10^{-1}

Thus: x−2=0.1x - 2 = 0.1

Adding 2 to both sides: x=0.1+2=2.1x = 0.1 + 2 = 2.1

Therefore, rounded to 3 significant figures: x≈2.10x \approx 2.10

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