Simplify/Condense

Problem

3*ln(x3*y)+2*ln(y*z2)

Solution

1. Apply the power property of logarithms, which states n*ln(a)=ln(an) to move the coefficients into the exponents of the arguments.

ln((x3*y)3)+ln((y*z2)2)

2. Simplify the exponents inside the natural logs by applying the power of a product rule (a*b)n=an*bn

ln(x9*y3)+ln(y2*z4)

3. Apply the product property of logarithms, which states ln(a)+ln(b)=ln(a*b) to combine the two terms into a single natural log.

ln(x9*y3⋅y2*z4)

4. Simplify the expression inside the logarithm by adding the exponents of the common base y

ln(x9*y5*z4)

Final Answer

3*ln(x3*y)+2*ln(y*z2)=ln(x9*y5*z4)

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