Logs have some very useful properties and let's learn about them in A-Level Maths!
Properties
logb mn = logb m + logb n
logb (m/n) = logb m −logb n
logb ma = a logb m
logb m = logb n if and only if m = n
Note that for all of the above properties we require that b > 0, b ≠ 1, and m, n > 0.
Note also that logb 1 = 0 for any b ≠ 0 since b0 = 1.
In addition, logb b = 1 since b1 = b. We can apply these properties to simplify logarithmic expressions.
Example 1 :
logb (xy/z)
= logb xy −logb z
= logb x + logb y −logb z
Example 2 :
log5 5p = p log5 5
= p ×1
= p
Example 3 :
log2 (8x)1/3
= 1/3 log2 8x
= 1/3 [log2 8 + log2 x]
= 1/3 [3 + log2 x]
= 1 + 1/3 log2 x
Example 4 :
Find x if 2 logb 5 + 1/2 logb 9 −logb 3 = logb x
logb 52 + logb 91/2 −logb3 = logb x
logb 25 + logb 3 −logb 3 = logb x
logb 25 = logb x
x = 25
Example 5 :
log2 (8x3/2y)
= log2 8x3 −log2 2y
= log2 8 + log2 x3 −[log2 2 + log2 y]
= 3 + 3 log2 x −[1 + log2 y]
= 3 + 3 log2 x −1 −log2 y
= 2 + 3 log2 x −log2 y
Exercises:
1. Use the logarithm laws to simplify the following:
(a) log2 xy −log2 x2
(b) log2 (8x2/y) + log2 2xy
(c) log3 9xy2 −log3 27xy
(d) log4(xy)3 −log4 xy
(e) log3 9x4 −log3(3x)2
2. Find x if:
(a) 2 logb 4 + logb 5 −logb 10 = logb x
(b) logb 30 −logb 52 = logb x
(c) logb 8 + logb x2 = logb x
(d) logb(x + 2) −logb 4 = logb 3x
(e) logb(x −1) + logb 3 = logb x
Ans:
1. (a) log2 (y/x)
(b) 4 + 3 log2 x
(c) log3 y −1
(d) 2 log4(xy)
(e) 0
2. (a) 8
(b) 6/5
(c) 1/8
(d) 2/11
(e) 1.5
Drafted by Eunice (Maths)