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, .

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)