1.14: Logarithms in terms of p and q; exponential equation

0606/13/O/N/19 — Question 8 · 8 marks · Log laws, Exponentials, Equations

(a) Given that log⁡ax=p\log_{a} x = p and log⁡ay=q\log_{a} y = q, find, in terms of pp and qq:
(a)(i)
log⁡aaxy2\log_{a} axy^{2}
[2]
(a)(ii)
log⁡a(x3ay)\log_{a}\left(\dfrac{x^{3}}{ay}\right)
[2]
(a)(iii)
log⁡xa+log⁡ya\log_{x} a + \log_{y} a
[1]
(b)
Using the substitution m=3xm = 3^{x}, or otherwise, solve 3x−31+2x+4=03^{x} - 3^{1+2x} + 4 = 0.
[3]