Without using calculator or tables, simplify log7root7-log5root5 divided by log 1.4
(log(7√7)-log(5√5))/log(1.4)
3/2 (log7 - log5)/log 1.4
3/2 log 7/5 / log 1.4
But, 7/5 = 1.4, so we wind up with just
3/2
To simplify the expression log7(root7) - log5(root5) divided by log 1.4 without using a calculator or tables, we can use the properties of logarithms and basic algebraic manipulations.
First, let's simplify the numerator:
log7(root7) - log5(root5)
Using the property of logarithms that states log a(b) - log a(c) = log a(b/c), we can combine the logarithms:
log7(root7/root5)
Now, let's simplify the denominator:
log 1.4
Since log 1.4 is not a commonly known value, we can use a basic logarithmic identity to express it in terms of more familiar logarithms:
log a(b) = log c(b) / log c(a)
Using this identity, we can rewrite log 1.4:
log 1.4 = log 10(1.4) / log 10(10)
The base 10 logarithm of 1.4 is not immediately apparent, but we can approximate it by understanding that log values increase by 0.3 for each power of 2, since 2^0.3 ≈ 1.26. So, we know that log 10(1.26) ≈ 0.1.
Now, we can rewrite log 1.4:
log 1.4 ≈ 0.1 / 1
Since dividing by 1 does not change the value, we have:
log 1.4 ≈ 0.1
So, our simplified expression becomes:
log7(root7/root5) / log 1.4 ≈ log7(root7/root5) / 0.1
At this point, we cannot simplify any further without using a calculator or tables.