Without using maths table or calc,simplify:log(4/9)/log(27/8) +1

Answer 1

To simplify the expression without using a math table or calculator, we can use the properties of logarithms and some basic knowledge of fractions.

First, let's simplify the fractions inside the logarithms by finding their common bases:

4/9 = (2^2)/(3^2) = 2^2 * (3^2)^(-1) = 2^2 * 3^(-2)

27/8 = (3^3)/(2^3) = 3^3 * (2^3)^(-1) = 3^3 * 2^(-3)

Now, let's rewrite the logarithmic expression using the properties of logarithms (specifically the change of base formula):

log(4/9)/log(27/8) + 1
= (log(2^2 * 3^(-2))) / (log(3^3 * 2^(-3))) + 1
= (2 * log(2) - 2 * log(3)) / (3 * log(3) - 3 * log(2)) + 1

Since we are not allowed to use math tables or calculators, we cannot further simplify the expression. Therefore, the simplified form of log(4/9)/log(27/8) + 1 is:

(2 * log(2) - 2 * log(3)) / (3 * log(3) - 3 * log(2)) + 1