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4. 3 + log a = log (b + 4) 10. log(6x – 19x + 5) – 2 log x = 1
2
2 8
1 log(6x − 19x + 5) − log x = log 10
2
2
log 2 + log a = log (b + 4) 3
3
2 2 2 6x – 19x + 5
2
1 = 10
8a = (b + 4) 3 x 2
1 (4x − 1)(x + 5) = 0
a = (b + 4) 3 x = −5 (rejected) or x = 1
8 [4] 4
[5]
log √x
5. log y – 3 = log 3

4
3
3 log 3 2 3 11. log (xy ) = 10
a
3
y log x + 3 log y = 10
a
a
1 = 3 3 2
4
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log (x y ) = 16
(√x) 2 1 3 log x + 2 log y = 16
a
y = 81x 4 a a
[5] Solve the simultaneous equations to get: = 1
log y = 2 2
a
1 log x = 4
6. log (x + y) − log 2 = (log xy) a = √3
2
a a a log √xy
2
x + y = (xy) 1 2 1 a = √(7) –
2 = [log x + log y]
2
a
a
1 x + y 2 2 = xy = [4 + 2] = 1 8 + 3
1
2
2
2
x + 2xy + y = 4xy = 3 2 2
2
2
1
x + y = 2xy (proven) [7] = 2 2
2
2
[4] =
2
3
log y 12. log x – log y = log xy + log 2
2
2
2
2
7. log x + 2 log 3 2 = log 3 x 3 = (xy)(2 )
3
–2
2
3
3
3
log x + log y = log 3 −2 y y = ± x
3
3
3
xy = 3 −2 2 [5]
= 1
9 [3] 13. (x − 4)e – 4(x − 4) = 0
2
3 −
x
2
2
2
3 −
4
3
x
2
8. (a) log c (x − 4) e – 4 = 0
3
ab 2
log c
3 −
x
= 3 3 log a + log b 4 x − 4 = 0 or e – 4 = 0
2
c
2
c
c
x
1
= 3 1 x + y 2 x = ±2 or 3 − = ln 4
x = 1.24

= 3 [3]
x + y [3] 14. (a) log p 3
ab
3 log p
a1 2
(b) log b – log 1 c = log a + log b
p
a
log b p p
= c – log c = 3
–1
log a a x + y [3]
c
= y + 1
x x ab 3
y + 1 (b) log a 1 p 2
=
x
[3] log p 1 ab 3 2
= p
9. log x + 2 log y = 4 log a
3 9 p
log y 2 log a + log b – log p
3
log x + 3 = log 3 = p p p
4
3 log 3 2 3 log a
3
log (xy) = log 3 x + 3y – 1 p
4
3 xy = 81 3 = x
[3] [3]
Answers 179
Answers Add Math.indd 179 14/03/2022 12:29 PM