Question
Write \(\ln ({x^2} – 1) – 2\ln (x + 1) + \ln ({x^2} + x)\) as a single logarithm, in its simplest form.
▶️Answer/Explanation
Markscheme
\(\ln ({x^2} – 1) – \ln {(x + 1)^2} + \ln x(x + 1)\) (A1)
\( = \ln \frac{{x({x^2} – 1)(x + 1)}}{{{{(x + 1)}^2}}}\) (M1)A1
\( = \ln \frac{{x(x + 1)(x – 1)(x + 1)}}{{{{(x + 1)}^2}}}\) (A1)
\( = \ln x(x – 1)\,\,\,\,\,\left( { = \ln ({x^2} – x)} \right)\) A1
[5 marks]
Question
Let \(g(x) = {\log _5}\left| {2{{\log }_3}x} \right|\) . Find the product of the zeros of g .
▶️Answer/Explanation
Markscheme
\(g(x) = 0\)
\({\log _5}\left| {2{{\log }_3}x} \right| = 0\) (M1)
\(\left| {2{{\log }_3}x} \right| = 1\) A1
\({\log _3}x = \pm \frac{1}{2}\) (A1)
\(x = {3^{ \pm \frac{1}{2}}}\) A1
so the product of the zeros of g is \({3^{\frac{1}{2}}} \times {3^{ – \frac{1}{2}}} = 1\) A1 N0
[5 marks]
Question
a.State the set of values of \(a\) for which the function \(x \mapsto {\log _a}x\) exists, for all \(x \in {\mathbb{R}^ + }\).[2]
b. Given that \({\log _x}y = 4{\log _y}x\), find all the possible expressions of \(y\) as a function of \(x\).[6]
▶️Answer/Explanation
Markscheme
a. \(a > 0\) A1
\(a \ne 0\) A1
[2 marks]
METHOD 1
\({\log _x}y = \frac{{\ln y}}{{\ln x}}\) and \({\log _y}x = \frac{{\ln x}}{{\ln y}}\) M1A1
Note: Use of any base is permissible here, not just “e”.
\({\left( {\frac{{\ln y}}{{\ln x}}} \right)^2} = 4\) A1
\(\ln y = \pm 2\ln x\) A1
\(y = {x^2}\;\;\;\)or\(\;\;\;\frac{1}{{{x^2}}}\) A1A1
METHOD 2
\({\log _y}x = \frac{{{{\log }_x}x}}{{{{\log }_x}y}} = \frac{1}{{{{\log }_x}y}}\) M1A1
\({({\log _x}y)^2} = 4\) A1
\({\log _x}y = \pm 2\) A1
\(y = {x^2}\;\;\;\)or\(\;\;\;y = \frac{1}{{{x^2}}}\) A1A1
Note: The final two A marks are independent of the one coming before.
[6 marks]
Total [8 marks]
Question
(a) Let a = log2x, b = log2y, c = log2z. Write \(log_2(\frac{x^3\sqrt{y}}{z^4})\) in terms of a, b and c.
(b) Let A = log2x, B = log4y, C = log8z. Write \(log_2(\frac{x^3\sqrt{y}}{z^4})\) in terms of A, B and C.
▶️Answer/Explanation
Ans
(a) \(log_2(\frac{x^3\sqrt{y}}{z^4})=3a+\frac{1}{2}b-4c\).
(b) \(log_2(\frac{x^3\sqrt{y}}{z^4})\)=3A+B-12C\)
Question
Solve the simultaneous equation \(2^{x^2}=4^{y}\) and \(log_zy=\frac{3}{2}\)
▶️Answer/Explanation
Ans
x = 4, y = 8