Without using maths table or calc,simplify:log(4/9)/log(27/8) +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
To simplify the expression log(4/9)/log(27/8) + 1 without using a math table or calculator, we can start by simplifying each logarithm separately.
First, let's simplify log(4/9).
We can rewrite 4/9 as a power of a common base. Since logarithms are typically taken with base 10, we can rewrite 4/9 as (2^2) / (3^2).
So, log(4/9) becomes log[(2^2) / (3^2)].
According to the properties of logarithms, we can rewrite this expression as log(2^2) - log(3^2).
Using another property of logarithms, we can simplify this further as 2 log(2) - 2 log(3).
Next, let's simplify log(27/8).
Similar to the previous step, we can rewrite 27/8 as (3^3) / (2^3).
So, log(27/8) becomes log[(3^3) / (2^3)].
Using the properties of logarithms again, we can rewrite this expression as log(3^3) - log(2^3).
Simplifying further, we get 3 log(3) - 3 log(2).
Now, substitute these simplified expressions back into the original expression:
[2 log(2) - 2 log(3)] / [3 log(3) - 3 log(2)] + 1.
To simplify further, we can combine like terms:
[2 - 2 log(3)/log(2)] / [3 - 3 log(2)/log(3)] + 1.
And that's the simplified expression without using a math table or calculator!
To simplify the expression log(4/9)/log(27/8) + 1, we can start by simplifying each logarithm individually.
Let's simplify the logarithm log(4/9) first. We know that log(a/b) is equal to log(a) - log(b). So, we can rewrite log(4/9) as log(4) - log(9).
Next, let's simplify the logarithm log(27/8). Similarly, we can rewrite it as log(27) - log(8).
Now, the expression becomes (log(4) - log(9))/(log(27) - log(8)) + 1.
To simplify further, we can combine the logarithms with the same base. log(4) - log(9) can be rewritten as log(4/9), and log(27) - log(8) as log(27/8). The expression now becomes log(4/9)/log(27/8) + 1.
So, the simplified expression is log(4/9)/log(27/8) + 1.