1. Logarithms to an Arbitrary Base

2. I learnt that
if x = 10y, then y = log x.
How about if x = ay (where a > 0 and a  1)? Can we find the value of y? Is y the logarithm of x?

3. If x = ay, then y = loga x, where a > 0 and a  1.
When x is expressed as the base a,
x = ay
(where a > 0 and a  1)
the index y is called the logarithm of x to the base a.
Symbolically:
y = loga x
So, we have:
If x = ay, then y = loga x, where a > 0 and a  1.

4. x = ay > 0 for all values of y
For example:
8 = 23
log2 8 = 3
36 = 62
log6 36 = 2
Note: loga x is undefined for x  0.
x = ay > 0 for all values of y
e.g. log4 0 and log3 (3) are undefined.

5. Exponential form (x = ay) Logarithmic form (y = loga x)1
7 = 71
◄ log7 71 = 1
log2 1 =
1 = 20
◄ log2 20 = 0
= 51 ø ö ç è æ 5 1
log = 1
◄ log5 5–1 = –1
= 63 ø ö ç è æ
216 1
log 6 = 3
◄ log6 6–3 = –3
Similar to common logarithm, we notice that:
loga a = 1 and loga 1 = 0,
where a > 0 and a  1

6. In fact, the following is also true:
If y = loga x, then x = ay,
where a > 0 and a  1.
For example:
log4 x = 3
x = 43
= 64
log9 x = 2 1
x = 9 2 1
= 3

7. Follow-up question
Find the values of the following logarithms without using a calculator.
(a) log3 27
∵ 27 = 33
◄ Express 27 as a power of the base 3.
∴ log3 27 = 3
◄ If x = ay, then loga x = y.
(b) ø ö ç è æ 49 1
log 7 49 1 7 2 = - ∵
◄ Express as a power of the base 7. 49 1 2 - = ∴ ø ö ç è æ 49 1
log 7

8. base-change formula of logarithms.
Properties of Logarithms to an Arbitrary Base
Let a > 0 and a  1. For any M, N > 0, we have:
1. loga (MN) = loga M + loga N
2. loga = loga M  loga N
3. loga Mn = n loga M
◄ n is a real number. a N b
log = 4.
◄ b > 0 and b  1
This is called the
base-change formula of logarithms.

9. Properties of Logarithms to an Arbitrary Base
Let a > 0 and a  1. For any M, N > 0, we have:
1. loga (MN) = loga M + loga N
2. loga = loga M  loga N
3. loga Mn = n loga M
◄ n is a real number. a N b
log = 4.
◄ b > 0 and b  1
The proof for Property (4) can be viewed in Proof of Logarithms.ppt.

10. Take b = 10 in the base-change formula
By using the base-change formula, you can find the values of logarithms to an arbitrary base with a calculator.
For example, to find the value of log2 5,
Take b = 10 in the base-change formula .

11. For example, to find the value of log2 5,
By using the base-change formula, you can find the values of logarithms to an arbitrary base with a calculator.
For example, to find the value of log2 5,
◄ Key in

12. Can you find the value of log6 72 - log6 2 without using a calculator?ø ö ç è æ = 2 72
log 6
= log6 36
= log6 62
= 2
by the definition of logarithms

13. Follow-up question
Find the value of log4 8 without using a calculator. 4
log 8 2 = ◄ = N a
log b
log4 8 2
log 3 = 2 3 =
◄ By the definition of logarithms.

14. For logarithmic equations to an arbitrary base, like log5 (x  3) = 2 and log3 (4x) = log3 8, you can solve them as follows.
1. Use the definition and properties of logarithms.
2. Apply the following property:
If loga x = loga y,
then x = y.

15. If loga x = loga y, then x = y.
For example,
(a) ∵
log5 (x  3) = 2
If loga y = 2,
then y = a2. ∴
x  3 = 52
x  3 = 25
x = 28
◄ Checking:
L.H.S. = log5 (28  3) = log5 25 = log5 52 = 2
= R.H.S.
(b) ∵
log3 (4x) = log3 8
If loga x = loga y, then x = y. ∴
4x = 8
x = 2
◄ Checking:
L.H.S. = log3 [4(2)] = log3 8 = R.H.S.

16. How about if the bases of two logarithms are different?
You can apply the base-change formula first.

17. Follow-up question Solve . ◄ Apply the base-change formula.
◄ log2 4 = log2 22 = 2
◄ n loga M = loga Mn