Write the expression as the logarithm of a single quantity

Question

3[(ln x) -2ln(x^2+1)]+2 ln 5?

Answer 1

ln{5^2 * [x / (x^2 + 1)^2]^3}

Answer 2

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 ]