evaluate without using a calculator or mathematical table:log32+log128-log729/log32+log2-log27

Answer 1

To evaluate the expression without a calculator or mathematical table, we can use the properties of logarithms.

First, let's simplify each logarithm:

1. log32 = log(2^5) = 5log2
2. log128 = log(2^7) = 7log2
3. log729 = log(3^6) = 6log3
4. log27 = log(3^3) = 3log3
5. log2 = log(2^1) = log2

Now, substitute these simplified logarithms back into the expression:

log32 + log128 - log729 / log32 + log2 - log27
= 5log2 + 7log2 - 6log3 / 5log2 + log2 - 3log3

Using the properties of logarithms, we can combine the logarithms with the same base:

= (5 + 7)log2 - (6 + 3)log3 / (5 + 1)log2 - 3log3
= 12log2 - 9log3 / 6log2 - 3log3

Now, let's simplify the expression further:

= 3(4log2 - 3log3) / 3(2log2 - log3)
= 4log2 - 3log3 / 2log2 - log3

Since the denominators are the same, we can treat this expression as a fraction:

= (4log2 - 3log3) / (2log2 - log3)

Now, let's evaluate the expression without a calculator or mathematical table:

We can rewrite each logarithm using the change of base rule:

= (4log(10)/log(2) - 3log(10)/log(3)) / (2log(10)/log(2) - log(10)/log(3))

Since log(10) is a commonly known logarithm value, we can substitute it with its approximate value of 1:

= (4(1)/log(2) - 3(1)/log(3)) / (2(1)/log(2) - (1)/log(3))
= (4/log(2) - 3/log(3)) / (2/log(2) - 1/log(3))

Now, we need to approximate the logarithmic values:

- log(2) is approximately 0.301
- log(3) is approximately 0.477

Substituting these values into the expression:

= (4/0.301 - 3/0.477) / (2/0.301 - 1/0.477)
= (13.29 - 6.29) / (6.63 - 2.09)
= 7 / 4.54
= 1.541

Therefore, the approximate value of the expression "log32+log128-log729/log32+log2-log27" is approximately 1.541.