Write the expression as the logarithm of a single quantity
3[(ln x) -2ln(x^2+1)]+2 ln 5?
ln ( a ^ n ) = n * ln a
So:
2 ln 5 = ln ( 5 ^ 2 ) = ln 25
Now:
3 [ ( ln x ) - 2 ln ( x ^ 2 + 1 ) ] + 2 ln 5 =
3 [ ( ln x ) - 2 ln ( x ^ 2 + 1 ) ] + ln 25 =
3 * ( ln x ) - 3 * 2 ln ( x ^ 2 + 1 ) + ln 25 =
3 ( ln x ) - 6 ln ( x ^ 2 + 1 ) + ln 25
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Remark:
ln ( a ^ n ) = n * ln a
3 ( ln x ) = ln ( x ^ 3 )
6 ln ( x ^ 2 + 1 ) = ln ( x ^ 2 + 1 ) ^ 6
ln ( a * b ) = ln a + ln b
ln ( x ^ 3 ) + ln 25 = ln ( 25 * x ^ 3 ) = ln ( 25 x ^ 3 )
ln ( a / b) = ln a - ln b
ln ( x ^ 3 ) + ln 25 - ln [ ( x ^ 2 + 1 ) ^ 6 ] = ln [ 25 x ^ 3 / ( x ^ 2 + 1 ) ^ 6 ]
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3 [ ( ln x ) - 2 ln ( x ^ 2 + 1 ) ] + 2 ln 5 = ln [ 25 x ^ 3 / ( x ^ 2 + 1 ) ^ 6 ]
ln{5^2 * [x / (x^2 + 1)^2]^3}
To express the given expression as the logarithm of a single quantity, we can use the properties of logarithms. Let's simplify step-by-step:
1. Use the power rule of logarithms:
ln(a^b) = b * ln(a)
2. Simplify the expression inside the parentheses:
-2ln(x^2+1)
= ln((x^2+1)^-2)
= ln(1/(x^2+1)^2)
3. Using the product rule of logarithms, we can separate the terms:
[(ln x) -2ln(x^2+1)]
= ln(x) + ln(1/(x^2+1)^2)
4. Distribute the coefficient 3 to each term:
3[(ln x) -2ln(x^2+1)]
= 3ln(x) + 3ln(1/(x^2+1)^2)
5. Combine the two terms by using the sum rule of logarithms:
3ln(x) + 3ln(1/(x^2+1)^2) + 2ln 5
= ln(x^3) + ln(1/(x^2+1)^2) + ln(5^2)
6. Combine the terms inside the parentheses into a single logarithm using the power rule:
ln(x^3) + ln(1/(x^2+1)^2) + ln(5^2)
= ln(x^3 * 1/(x^2+1)^2 * 5^2)
7. Simplify the expression inside the logarithm:
ln(x^3 * 1/(x^2+1)^2 * 5^2)
= ln(5^2 * x^3 / (x^2+1)^2)
Therefore, the given expression can be expressed as ln(5^2 * x^3 / (x^2+1)^2).
To write the given expression as the logarithm of a single quantity, we need to simplify it by using some logarithmic properties.
Let's break down the expression step by step:
1. First, distribute the 3 to the terms inside the brackets:
3(ln x) - 6ln(x^2 + 1)
2. Next, using the logarithmic property, we can combine the two terms:
ln(x^3) - ln((x^2 + 1)^6)
3. Now, applying another logarithmic property, we can simplify further:
ln(x^3/(x^2 + 1)^6)
4. Finally, simplify the expression inside the logarithm by simplifying the numerator and denominator:
ln(x^3/(x^2 + 1)^6)
Therefore, the provided expression can be written as the logarithm of a single quantity: ln(x^3/(x^2 + 1)^6).