Question

Use logarithmic properties and the fact that ln(2) ≈ 0.69 and ln(3) ≈ 1.10 to approximate the value of each of the

following logarithmic expressions. Do not use a calculator.
ln(108)

Answer 1

ln(108) = 3.58

Explanation:

Data provided:

ln(2) ≈ 0.69

and

ln(3) ≈ 1.10

equation to be solved

ln(108)

now,

ln(108) = ln (2 × 54)

or

= ln( 2 × 2 × 27 )

or

= ln( 2 × 2 × 3 × 3 × 3 )

or

= ln( 2² × 3³ )

Now,

we have the property of log function as:

1) ln(AB) = ln(A) + ln(B)

and,

2) ln(xⁿ) = n × ln(x)

on applying the above properties , we get

ln( 2² × 3³ ) = ln(2²) + ln(3³) using property (1)

or

= 2ln(2) + 3ln(3) using property (2)

on substituting the values given in the question

we get

ln(108) = ( 2 × 0.69 ) + ( 2 × 1.10 )

= 1.38 + 2.20

= 3.58