Log5(3÷5)+3log5(15÷2)-log5(81÷8)
assuming base 5 for all the logs, we have
(log3-log5) + 3(log15-log2) - (log81-log8)
log3-log5 + 3(log3+log5-log2) - (4log3-3log2)
log3-log5+3log3+3log5-3log2-4log3+3log2
2log5
= 2
The solving are too confusing it should show more steps for Better understanding.thanks it was helpful
Its kind of confusing...
It should be explained better
I to write wassce i need help
Intelligent
To simplify the expression, we will use the following logarithmic rules:
1. log(a/b) = log(a) - log(b)
2. log(a^n) = n*log(a)
Let's simplify step by step:
1. Start with the expression:
log5(3÷5) + 3log5(15÷2) - log5(81÷8)
2. Apply the rule log(a/b) = log(a) - log(b) to simplify each term within the parentheses:
log5(3) - log5(5) + 3(log5(15) - log5(2)) - log5(81) + log5(8)
3. Apply the rule log(a^n) = n*log(a) to further simplify:
log5(3) - log5(5) + 3(log5(15) - log5(2)) - log5(81) + 3(log5(8))
4. Simplify further, using basic logarithmic properties:
log5(3) - log5(5) + 3log5(15) - 3log5(2) - log5(81) + 3log5(8)
5. Combine like terms within the expression:
[log5(3) + 3log5(15) - log5(81)] - [log5(5) + 3log5(2) - 3log5(8)]
6. Use the logarithmic rules one more time:
[log5(3*15) - log5(81)] - [log5(5) + log5(2^3) - log5(8^3)]
7. Simplify the expressions within the logarithms:
[log5(45) - log5(81)] - [log5(5) + log5(8) - log5(8)]
8. Apply the rule log(a/b) = log(a) - log(b) one more time:
log5(45/81) - [log5(5) + log5(8/8)]
9. Simplify the expressions within the logarithms:
log5(5/9) - [log5(5) + log5(1)]
10. Simplify further:
log5(5/9) - [log5(5) + 0]
11. Combine like terms:
log5(5/9) - log5(5)
12. Apply the rule log(a/b) = log(a) - log(b):
log5(5/9 * 1/5)
13. Simplify the expression within the logarithm:
log5(1/9)
14. Simplify the expression:
log5(1) - log5(9)
15. Simplify further:
0 - log5(9)
16. Finally, evaluate the logarithm:
-log5(9)
So, the simplified expression is -log5(9).
For any base of logarithm:
log ( a ∙ b ) = log ( a ) + log ( b )
log ( a ÷ b ) = log ( a ) - log ( b )
log ( aᵐ ) = m ∙ log ( a )
In this case:
log5 (3÷5) = log5 ( 3 ) - log5 ( 5 ) = log5 ( 3 ) - 1
log5 (15 ) = log5 ( 3 ∙ 5 ) = log5 ( 3 ) + 1
log5 (15÷2 ) = log5 ( 15 ) - log5 ( 2 ) = log5 ( 3 ) + 1 - log5 ( 2 )
__________
becouse:
log5 ( 5 ) = 1
__________
log5 ( 81 ) = log5 (3⁴ ) = 4 ∙ log5 ( 3 )
log5 ( 8 ) = log5 (2³ ) = 3 ∙ log5 ( 2 )
log5 ( 81÷8 ) = log5 ( 81 ) - log5 ( 8 ) = 4 ∙ log5 ( 3 ) - 3 ∙ log5 ( 2 )
So:
log5(3÷5)+3log5(15÷2)-log5(81÷8) =
log5 ( 3 ) - 1 + 3 ∙ [ log5 ( 3 ) + 1 - log5 ( 2 ) ] - [ 4 ∙ log5 ( 3 ) - 3 ∙ log5 ( 2 ) ] =
log5 ( 3 ) - 1 + 3 log5 ( 3 ) + 3 - 3 log5 ( 2 ) - 4 log5 ( 3 ) + 3 log5 ( 2 ) =
4 ∙ 3 log5 ( 3 ) - 4 ∙ 3 log5 ( 3 ) + 2 - 3 ∙ log5 ( 2 ) + 3 ∙ log5 ( 2 ) = 2