Log5(3/5)+3log5(15/2)-log5(81/8)
Log5(3/5)+3log5(15/2)-log5(81/8)
= Log5(3/5)+log5((15/2)^3)-log5(81/8)
= log5[ (3/5)(3375/8) รท (81/8) ]
= log5 [25]
= 2
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Well, let me put on my mathematical clown nose and crunch some numbers for you!
Log5(3/5) can be simplified by applying the power rule of logarithms:
Log5(3) - Log5(5) = Log5(3) - 1
So, we have:
Log5(3) - 1 + 3(Log5(15) - Log5(2)) - Log5(81) + Log5(8)
Now, let's simplify further using the power rule again:
Log5(3) - 1 + 3(Log5(15) - Log5(2)) - Log5(81) + 3(Log5(8) - 1)
Now, we can substitute the values into each logarithm:
Log5(3) - 1 + 3(Log5(3 * 5) - Log5(2)) - Log5(3^4) + 3(Log5(2^3) - 1)
Log5(3) - 1 + 3(Log5(3) + Log5(5) - Log5(2)) - Log5(81) + 3(Log5(2) + Log5(2^2) - 1)
Log5(3) - 1 + 3(Log5(3) + 1 - Log5(2)) - Log5(81) + 3(Log5(2) + 2 - 1)
Now, let's simplify each term:
Log5(3) - 1 + 3Log5(3) + 3 - 3Log5(2) - Log5(81) + 3Log5(2) + 6 - 3
And finally, we can combine like terms:
4Log5(3) - Log5(81) + 2
And there you have it! The simplified expression is 4Log5(3) - Log5(81) + 2. Hope that brings a smile to your face!
To simplify the given expression, we can use the properties of logarithms. Let's break down the expression step by step:
1. Logarithm of a quotient:
log base b (a / c) = log base b (a) - log base b (c)
Using this property, we can simplify the first term: log5(3/5) becomes log5(3) - log5(5).
2. Logarithm of a product:
log base b (a * c) = log base b (a) + log base b (c)
Using this property, we can simplify the second term: 3log5(15/2) becomes 3(log5(15) + log5(2)).
3. Exponent property of logarithms:
log base b (x^a) = a * log base b (x)
Using this property, we can simplify the third term: log5(81/8) becomes log5(81) - log5(8).
Now let's substitute the simplified expressions back into the original expression:
log5(3/5) + 3log5(15/2) - log5(81/8)
= (log5(3) - log5(5)) + 3((log5(15) + log5(2)) - (log5(81) - log5(8)))
Next, let's apply the properties of logarithms and simplify further:
= log5(3) - log5(5) + 3log5(15) + 3log5(2) - 3log5(81) + 3log5(8)
Now we can evaluate each term using the definition of logarithm:
= log5(3) - log5(5) + 3 * (log5(15) + log5(2)) - 3 * (log5(81) - log5(8))
Finally, we compute the logarithms of the given numbers:
= log5(3) - log5(5) + 3 * (log5(15) + log5(2)) - 3 * (log5(81) - log5(8))
= log5(3) - log5(5) + 3 * (log5(15) + log5(2)) - 3 * (4 - 3)
Since log5(81) = 4 and log5(8) = 3:
= log5(3) - log5(5) + 3 * (log5(15) + log5(2)) - 3 * (4 - 3)
= log5(3) - log5(5) + 3 * (log5(15) + log5(2)) - 3
Now, this expression cannot be simplified further unless we have specific values for log5(3), log5(5), log5(15), and log5(2).
assuming all logs have base 5, we have
log(3/5)+3log(15/2)-log(81/8)
= log3 - log5 + 3(log3+log5-log2) - 4log3 + 3log2
= log3 - log5 + 3log3 + 3log5 - 3log2 - 4log3 + 3log2
= 2log5
and since log5(5) = 1, the final value is 2, as above