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 \(a\).