Logarithms to an Arbitrary Base

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?

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.

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.

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

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

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

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.

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.

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 .

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 5 2 .

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

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.

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.

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.

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

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