© John Wiley & Sons Australia, Ltd Page 1
WorkSHEET 18.4 Real numbers IV Name: ___________________________ 1 Evaluate log5 .
log! √5 = log! 5"#
=12 log! 5
=12 × 1
=12
Or use the Sock rule.
2 Evaluate log! 25 log! 25
= log! 5#
= 2 log! 5
= 2 × 1
= 2 Or, use the Sock rule.
3 Solve for x:
= x
2x = 22 ´
2x =
x =
4 Solve for 𝑥; log$ 𝑥 = 3 Or use the Sock rule.
log$ 𝑥 = 3
𝑥 = 3$
𝑥 = 27
5 Solve for 𝑥; log!(𝑥 + 1) = 2 Or use the Sock rule.
𝑥 + 1 = 5#
𝑥 = 24
5
x=24log2 24log221
225
2
25
© John Wiley & Sons Australia, Ltd Page 2
6 Solve for x:
= 0.5
log3 x = 90.5 log3 x = 3 x = 33 x = 27
7 Solve for 𝑥; log! 𝑥 = log! 10 − log! 2
log! 𝑥 = log! 10 − log! 2
log! 𝑥 = log!102
log! 𝑥 = log! 5
log! 𝑥 = 1
Use the Sock rule to get;
𝑥 = 5"
𝑥 = 5
8 Simplify: (a) (b) (c) log# 16$(&'")
log) 7& = 𝑥 log) 7
= 𝑥 × 1
= 𝑥
log) 49& = log) 7#&
= 2𝑥 × 1
= 2𝑥
log# 16$(&'") = log#(2*)$(&'")
= log# 2"#(&'")
= 12(𝑥 + 1) log# 2
= 12𝑥 + 12
5.0)(loglog 39 =x )(loglog 39 x
x7log7
x49log7
© John Wiley & Sons Australia, Ltd Page 3
9 Solve for x:
loga x = loga (4 ´ 11) x = 44
10 Solve for y:
logb y = logb
y =
11 Solve for x:
6log5 x = 7log5 x - 3 -log5 x = -3 log5 x = 3 x = 53 x = 125
12 Solve for 𝑥: log# 𝑥 + log#(𝑥 + 2) = 3
log# 𝑥 + log#(𝑥 + 2) = 3
log# 𝑥(𝑥 + 2) = 3
Sock rule.
𝑥(𝑥 + 2) = 2$
𝑥# + 2𝑥 = 8
𝑥# + 2𝑥 − 8 = 0
(𝑥 + 4)(𝑥 − 2) = 0
𝑥 = −4𝑜𝑟2 Because log#−4 is undefined, there is only one solution,
𝑥 = 2
11log4loglog aaa x +=11log4loglog aaa x +=
5log13loglog bbb y -= 5log13loglog bbb y -=
÷øö
çèæ513
513
3log7)(log 56
5 -= xx 3log7)(log 56
5 -= xx
© John Wiley & Sons Australia, Ltd Page 4
13 Make x the subject of the following: (a) (b)
(a) (b) 3x = log10 y
14 Evaluate: 4 log! 82 log! 16
+ 3 log# √2!
4 log! 82 log! 16
+ 3 log# √2!
=4 log! 2$2 log! 2*
+ 3 log# 2"$
=12 log! 28 log! 2
+33 log# 2
=128 + 1
=32 + 1
= 212
15 Express b in terms of a given the following: (a)
(b)
(a)
(b)
xy 5=
xy 310=
xy 5=yx 5log=
xy 310=
3log10 yx =
ab =5log
ab =51log
ab 5=
ab51
=
© John Wiley & Sons Australia, Ltd Page 5
16 Solve for x, correct to 3 decimal places. (a) 3x = 8 (b) 8x = 75
(a) 3x = 8 log10 3x = log10 8 xlog10 3 = log10 8
x =
x 1.893 (b) 8x = 75 log10 8x = log10 75 xlog10 8 = log10 75
x =
x 2.076
3log8log
10
10
»
8log75log
10
10
»
