The curve \(C\) has polar equation
\(r=a \sin 3 \theta\)
where \(0 \leqslant \theta \leqslant \frac{1}{3} \pi\).
(i) Show that the area of the region enclosed by \(C\) is \(\frac{1}{12} \pi a^{2}\).
(ii) Show that, at the point of \(C\) at maximum distance from the initial line,
\(\tan 3 \theta+3 \tan \theta=0\)
(iii) Use the formula
\(\tan 3 \theta=\frac{3 \tan \theta-\tan ^{3} \theta}{1-3 \tan ^{2} \theta}\)
to find this maximum distance.
(iv) Draw a sketch of \(C\).
Solve the equation \(\frac{2}{\log _{x} 10}-\lg (x+4)=\lg 2\) for \(x\gt 0\).
(a) Write \(3+4 \log _{2} a-\log _{2} b\) as a single base 2 logarithm.
(b) Solve the equation \(\lg x=4 \log _{x} 10\).
Given that \(\log _{a}(p+1)+\frac{1}{\log _{p} a}-\log _{a}(p+2)+\log _{a} 5=\log _{a} 12\), find the value of \(p\).
Given that \(\log _{3} r+2 \log _{9} s=8\), find the value of \(r s\).
Find constants \(a\), \(b\) and \(c\) such that
\(\frac{\sqrt{p q^{4/3}}\,r^{-3}}{(pq^{-1})^2r^{-1}}=p^aq^br^c.\)
(a) (i) Find the set of values of \(x\) for which \(\lg(5x-3)\) exists.
(ii) Solve the equation \(\lg(5x-3)=1\).
(b) Given that \(y\gt 0\) and \(\log_y x=4+\frac12\log_y 64+\log_y 162\), find \(y\) in terms of \(x\), giving your answer in its simplest form.
(a) Write \(3+2\lg a-4\lg b\) as a single logarithm to base \(10\).
(b) Solve the equation \(3\log_a4+2\log_4a=7\).
(a) Solve the equation
\(\frac{9^{5x}}{27^{x-2}}=243.\)
(b) Given that
\(\log_a\sqrt b-\frac12=\log_b a,\)
where \(a\gt 0\) and \(b\gt 0\), solve this equation for \(b\), giving your answers in terms of \(a\).
Write
\(3\lg x+2-\lg y\)
as a single logarithm.
Given that \(\log_4 x=p\), find, in terms of \(p\),
(i) \(\log_4(16x)\),
(ii) \(\log_4\left(\dfrac{x^7}{256}\right)\).
(iii) Hence solve \(\log_4(16x)-\log_4\left(\dfrac{x^7}{256}\right)=5\), giving your answer correct to 2 decimal places.
(a) Solve \(\lg(x^2-3)=0\).
(b) (i) Show that \(\dfrac{\ln a^{\sin(2x+5)}+\ln\left(\frac1a\right)}{\ln a}\) can be written in the form \(\sin(2x+5)+k\), where \(k\) is an integer.
(ii) Hence find \(\displaystyle\int \dfrac{\ln a^{\sin(2x+5)}+\ln\left(\frac1a\right)}{\ln a}\,dx\).
(a) Given that \(\log_a x=p\) and \(\log_a y=q\), find, in terms of \(p\) and \(q\),
(i) \(\log_a axy^2\),
(ii) \(\log_a\left(\frac{x^3}{ay}\right)\),
(iii) \(\log_x a+\log_y a\).
(b) Using the substitution \(m=3^x\), or otherwise, solve
\(3^x-3^{1+2x}+4=0.\)
(a) Write
\((\log_2p)(\log_32)+\log_3q\)
as a single logarithm to base \(3\).
(b) Given that
\((\log_a5)^2-4\log_a5+3=0,\)
find the possible values of \(a\).
(a) Write
\((\log_2p)(\log_32)+\log_3q\)
as a single logarithm to base \(3\).
(b) Given that
\((\log_a5)^2-4\log_a5+3=0,\)
find the possible values of \(a\).
Solve the following equations.
(a) \(\log_5(5x-2)-\log_{25}x=\frac12\)
(b) \(\mathrm{e}^{3y-7}+\frac{4}{\mathrm{e}^3}=\frac{5}{\mathrm{e}^{3y-1}}\)
(a) Write \(5\lg a-4\lg b-3\) as a single base 10 logarithm.
(b) Solve the equation \(\log_5(x+1)-\log_{x+1}5=0\).
(a) Solve the equation \(\log_2x-4=5\log_x2\).
(b) Solve the equation \(\mathrm{e}^{x^2-3}=25\mathrm{e}^{7-x^2}\), giving your answers in exact form.
Solve the following equations. (a) \(\log _{2} x^{2}+\log _{16} x=18\) (b) \(\mathrm{e}^{2 x+1}-10 \mathrm{e}^{-2 x-1}=3\)
(a) Given that \(\log _{p} a+\log _{p} 12-\log _{p} 6=3 \log _{p} 4\), find the value of \(a\).
(b) Find the exact solutions of the equation \(4 \log _{3} x=9 \log _{x} 3\).