LOGARITHMS

Definition of a logarithm If a number y can be written in the form of ax, the index x is called the logarithm of y to the base a. In other word, if y = ax, then logay = x where y and a are positive numbers. Example 1: Write the exponent 3x = 16 into the logarithm form Example 2: Write the exponent y-5 = 4.6 into the logarithm form

Definition of a logarithm (cont) Example 3: Find the value of x in each of the following equations (a) log3x = 4 (b) log2x = -5 (c) log10(x – 4) = 3

Properties of Logarithm From the definition of logarithm, we can proceed further to obtain the following properties (a) loga1 = 0 (b) loga a = 1 (c) loga an = n

Properties of Logarithm (cont) Example 4: Without using the calculator, evaluate the following. log101 log2.5(5/2) log2 25 log381 log40.25 log8 2

The Rules of Logarithm Proof: Let loga x = m and loga y = n. So, x = am and y = an. Thus, xy = am an = a(m+n) Applying the definition of a logarithm gives

The Rules of Logarithm (cont) Proof: Let loga x = m and loga y = n. So, x = am and y = an. Thus, Applying the definition of a logarithm gives

The Rules of Logarithm (cont) Proof: Let loga x = m with x = am. Raising each side to the power r gives Applying the definition of a logarithm gives

The Rules of Logarithm (cont) Example 5: Simplify the following as a single logarithm

The Rules of Logarithm (cont) Example 6: Without using a calculator, evaluate the following.

The Rules of Logarithm (cont)

Natural Logarithm The most frequently used bases for log are 10 and the number ‘e’. Log to base 10 are known as common log Log to base e are called natural logarithm Use ‘ln’ to indicate ‘loge’ The rules of logarithm also apply to natural logarithm i.e

Natural Logarithm (cont) Example 7: Use the properties of logarithm to simplify the following expressions

Change of Base A logarithm of x with base a can be changed to base b and the formula is given as follows:

Solving equations involving indices Example 8: Given that 2.142x = 4.78, find the values of x, correct to four decimal places. Example 9: Solve 0.49x+1 = 0.62x.

Solve logarithm equations Example 10: Given that find the values of x. Example 11: Solve