# Simplify/Condense 1/9* natural log of (x+2)^9+1/2*( natural log of x- natural log of (x^2+3x+2)^2)

Simplify each term.
Simplify by moving inside the logarithm.
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Simplify.
Use the quotient property of logarithms, .
Simplify the denominator.
Factor using the AC method.
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Write the factored form using these integers.
Apply the product rule to .
Simplify by moving inside the logarithm.
Use the power rule to distribute the exponent.
Apply the product rule to .
Apply the product rule to .
Simplify the denominator.
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Simplify.
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Simplify.
Use the product property of logarithms, .
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify/Condense 1/9* natural log of (x+2)^9+1/2*( natural log of x- natural log of (x^2+3x+2)^2)