In IB Mathematics, The mathematics of logarithms and exponentials occurs naturally in many branches of science. It is very important in solving problems related to growth and decay. The growth and decay may be that of a plant or a population, a crystalline structure or money in the bank. Therefore we need to have some understanding of the way in which logs and exponentials work.

Definitions

If x and b are positive numbers and b =/= 1 then the logarithm of x to the base b is the power to which b must be raised to equal x. It is written logb x. In algebraic terms this means that

In IB Mathematics, The formula y = logb x is said to be written in logarithmic form and x = b^y is said to be written in exponential form. In working with these problems it is most important to remember that y = logb x and x = b^y are equivalent statements.

__Example 1 __:

If log_{4 }x = 2 then

x = 4^2

x = 16

__Example 2 __:

We have 25 = 5^2. Then log_{5} 25 = 2.

__Example 3 __:

If log_{9} x = 1/2 then

x = 9^ 1/2

x = √9

x = 3

Properties of Logs

In IB Mathematics, Logs have some very useful properties which follow from their definition and the equivalence of the logarithmic form and exponential form. Some useful properties are as follows:

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

In IB Mathematics, Note that for all of the above properties we require that b > 0, b =/= 1, and m, n > 0. Note also that log_{b }1 = 0 for any b>0 since b^0 = 1. In addition, log_{b} b = 1 since b^1 = 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 5^p= 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 9 ^1/2 − logb3 = logb x

logb 25 + logb 3 − logb 3 = logb x

logb 25 = logb x

x = 25

This is the end of this topic.