Math Problem Statement

3In(x+2)-In(x-1)

Solution

The expression you provided is:

3ln(x+2)ln(x1)3 \ln(x + 2) - \ln(x - 1)

To simplify this logarithmic expression, we can use the properties of logarithms. Specifically, the properties we'll use are:

  1. aln(b)=ln(ba)a \ln(b) = \ln(b^a) — logarithm power rule
  2. ln(a)ln(b)=ln(ab)\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) — logarithm subtraction rule

Step-by-step solution:

  1. Apply the power rule to the first term: 3ln(x+2)=ln((x+2)3)3 \ln(x + 2) = \ln((x + 2)^3)

    So the expression becomes: ln((x+2)3)ln(x1)\ln((x + 2)^3) - \ln(x - 1)

  2. Apply the logarithmic subtraction rule: ln((x+2)3x1)\ln\left(\frac{(x + 2)^3}{x - 1}\right)

Thus, the simplified form of the expression is: ln((x+2)3x1)\ln\left(\frac{(x + 2)^3}{x - 1}\right)

This is the final simplified expression.

Would you like more details on any step or have any questions?

Here are some related questions to explore:

  1. What is the derivative of this logarithmic expression?
  2. How do you simplify logarithmic expressions with more complex coefficients?
  3. What happens if x=1x = 1 in this expression?
  4. How does the logarithmic function behave when applied to negative inputs?
  5. Can this expression be expanded back to its original form?

Tip: Always remember to check the domain of logarithmic functions. Here, x>1x > 1, since both x+2x + 2 and x1x - 1 must be positive.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Properties of Logarithms
Simplification of Logarithmic Expressions

Formulas

a * ln(b) = ln(b^a) — Power rule of logarithms
ln(a) - ln(b) = ln(a/b) — Logarithmic subtraction rule

Theorems

Properties of Logarithms

Suitable Grade Level

Grades 10-12