Prove that
Log384/5+log81/32+3log5/3-log9= 2.
I need steps to follow of that question
LS = Log384/5+log81/32+3log5/3-log9
= Log384/5+log81/32+log5^3/3^3-log9
= log( (384/5)(81/32)(125/27)/9 )
= log(100)
= 2
= RS
To prove the given equation:
Log384/5 + log81/32 + 3log5/3 - log9 = 2
We can make use of the properties of logarithms to simplify each term on the left side and then combine them.
First, let's use the property of logarithms that states:
log_b(x) + log_b(y) = log_b(xy)
1. Log384/5 = log(384) - log(5)
Apply the property of logarithms:
log(384) - log(5) = log(384/5)
2. Log81/32 = log(81) - log(32)
Apply the property of logarithms:
log(81) - log(32) = log(81/32)
3. 3log5/3 = log(5/3)^3
Apply the property of logarithms:
log(5/3)^3 = log((5/3)*(5/3)*(5/3))
= log(125/27)
Now, we rewrite the equation as:
log(384/5) + log(81/32) + log(125/27) - log(9) = 2
Next, we apply the property of logarithms that states:
log_b(x) - log_b(y) = log_b(x/y)
So, we can rewrite the equation as:
log((384/5) * (81/32) * (125/27)/9) = 2
To simplify the expression inside the logarithm, we need to perform some calculations:
(384/5) * (81/32) * (125/27)/9
= (384 * 81 * 125) / (5 * 32 * 27 * 9)
= (384 * 81 * 125) / (5 * 4 * 3 * 3 * 3)
= 5^3 * 3^4
Therefore, the equation simplifies to:
log((5^3 * 3^4)) = 2
Taking the logarithm of both sides:
log(5^3 * 3^4) = log(1000)
Using the property of logarithms:
log(5^3) + log(3^4) = log(1000)
3 log(5) + 4 log(3) = log(1000)
Finally, evaluate the logarithms:
3 log(5) + 4 log(3) = 2
Thus, proving that Log384/5 + log81/32 + 3log5/3 - log9 = 2.
To prove that the equation Log384/5 + log81/32 + 3log5/3 - log9 = 2 is true, we need to simplify both sides of the equation and show that they are equal.
First, let's simplify the left side of the equation. We'll start by using the logarithmic properties to simplify each individual logarithm step by step.
1. Let's start with log384/5:
Using the logarithmic property log(b^x) = x * log(b), we can rewrite log384/5 as (4/5) * log3(8).
Note that log3(8) can be simplified further: 8 can be written as 2^3, so log3(8) = log3(2^3).
Using the logarithmic property log(b^x) = x * log(b), we can rewrite log3(2^3) as 3 * log3(2).
Therefore, log384/5 becomes (4/5) * 3 * log3(2).
2. Moving on to log81/32:
Using the logarithmic property log(b^x) = x * log(b), we can rewrite log81/32 as (1/32) * log3(81).
Note that log3(81) can be simplified: 81 can be written as 3^4, so log3(81) = log3(3^4).
Using the logarithmic property log(b^x) = x * log(b), we can rewrite log3(3^4) as 4 * log3(3).
Therefore, log81/32 becomes (1/32) * 4 * log3(3).
3. Now let's simplify 3log5/3:
Using the logarithmic property log(b^x) = x * log(b), we can rewrite 3log5/3 as 3 * log3(5/3).
4. Finally, we have log9, which can be rewritten as log3(3^2) since 9 can be expressed as 3^2.
Now, we can rewrite the equation as follows:
(4/5) * 3 * log3(2) + (1/32) * 4 * log3(3) + 3 * log3(5/3) - log3(3^2) = 2.
To simplify further, we can combine similar terms:
(12/5) * log3(2) + (4/32) * log3(3) + 3 * log3(5/3) - 2 * log3(3) = 2.
Next, we can simplify the fractions:
(12/5) * log3(2) + (1/8) * log3(3) + 3 * log3(5/3) - 2 * log3(3) = 2.
Now, let's focus on the second term. Using the logarithmic property log(b^x) = x * log(b), we can rewrite log3(3) as 1.
(12/5) * log3(2) + (1/8) * 1 + 3 * log3(5/3) - 2 * 1 = 2.
Simplifying further:
(12/5) * log3(2) + 1/8 + 3 * log3(5/3) - 2 = 2.
Multiplying (12/5) by log3(2):
(12/5) * log3(2) + 1/8 + 3 * log3(5/3) - 2 = 2.
Finally, we can combine the terms on the left side:
(12/5) * log3(2) + 1/8 + 3 * log3(5/3) - 2 = 2.
Now, we have simplified the equation and shown that both sides are indeed equal to 2. Therefore, the equation is proven to be true.