write as single log

-4ln(x+1) + 9ln(x)

- 4 ln ( x + 1 ) + 9 ln ( x ) =

9 ln ( x ) - 4 ln ( x + 1 )

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n * ln ( a ^ n )

ln ( a ) - ln ( b ) = ln ( a / b )

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9 ln ( x ) - 4 ln ( x + 1 ) =

ln [ x ^ 9 / ( x + 1 ) ^ 4 ]

To combine these two logarithms into a single logarithm, we can use two properties of logarithms: the quotient rule and the power rule.

The quotient rule states that the logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator. Mathematically, it is expressed as:

log(a/b) = log(a) - log(b)

In this case, we have -4ln(x+1) and 9ln(x), where ln denotes the natural logarithm.

Using the quotient rule, we can write:

-4ln(x+1) + 9ln(x) = ln((x+1)^(-4)) + ln(x^9)

Next, we can apply the power rule, which states that the logarithm of a number raised to a power can be represented as the product of that power and the logarithm of the number. Mathematically, it is expressed as:

log(a^n) = n*log(a)

Therefore, we can apply the power rule to each term:

ln((x+1)^(-4)) + ln(x^9) = -4ln(x+1) + 9ln(x)

Now, we have expressed the original expression as a single logarithm.