Note
Logarithms and exponentials
P2 2
Most calculators just have ‘log’ and not ‘log 10’ on their keys.
EXAMPLE 2.3
A ge metric se e ce begi s 2 1 The kth term is the first term in the sequence that is greater than 500 000. Find the value of k. SOLUTION
The kth term of a geometric sequence is given by ak = a
r k 1.
In this case a = 0.2 and r = 5, so: 0.2 × 5k −1 ! 500000 500000 0.2 ! 2500000
5k −1 !
5k −1
Taking logarithms to the base 10 of both sides: log10 5k 1 ! log10 2 500 000 (k
1)log10 5 ! log10 2 500 000 log10 2500 000 log10 5
k
1!
k
1 ! 9.15 k ! 10.15
Since k is an integer, then k = 11. So the 11th term is the first term greater than 500 000. Check :
510 511
10th term 0.2 11th term 0.2
1 1
390 625 (" 500 000) 1 953 125 (! 500 000)
Roots
A similar line of reasoning leads to the conclusion that: 1 n log x
log n x
The logic runs as follows: n
x
n
x
n
x
…
n
n log
n
x
x
x
log x
{
Since
n times it follows that and so 26
log n x
1 n log x