Math Problem Statement

Simplify the expression 4 ln(x) - (1/3) ln(x^2 + 1) + 2 ln(x - 1)

Solution

The expression you've provided is:

4ln(x)13ln(x2+1)+2ln(x1)4 \ln(x) - \frac{1}{3} \ln(x^2 + 1) + 2 \ln(x - 1)

To simplify this logarithmic expression, we can use the logarithmic properties:

  1. Multiplication Property: ln(a)+ln(b)=ln(ab)\ln(a) + \ln(b) = \ln(a \cdot b)
  2. Division Property: ln(a)ln(b)=ln(ab)\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)
  3. Power Property: nln(a)=ln(an)n \ln(a) = \ln(a^n)

Step-by-step simplification:

  1. Apply the power rule: 4ln(x)=ln(x4),2ln(x1)=ln((x1)2)4 \ln(x) = \ln(x^4), \quad 2 \ln(x - 1) = \ln((x - 1)^2)

    So, the expression becomes: ln(x4)13ln(x2+1)+ln((x1)2)\ln(x^4) - \frac{1}{3} \ln(x^2 + 1) + \ln((x - 1)^2)

  2. Combine the logarithmic terms using the addition and subtraction properties: ln(x4(x1)2(x2+1)1/3)\ln\left(\frac{x^4 \cdot (x - 1)^2}{(x^2 + 1)^{1/3}}\right)

Thus, the simplified form of the expression is: ln(x4(x1)2(x2+1)1/3)\ln\left(\frac{x^4 (x - 1)^2}{(x^2 + 1)^{1/3}}\right)

Would you like further clarification or details on any part of this process?

Here are 5 related questions:

  1. What are the domains of logarithmic functions like ln(x)\ln(x)?
  2. How do the properties of logarithms apply to real-world situations?
  3. Can logarithmic expressions always be simplified?
  4. How do you differentiate an expression involving logarithms?
  5. What happens when x=1x = 1 in the expression ln(x1)\ln(x - 1)?

Tip: Always check the domain restrictions of the variables inside logarithms, since logarithms are only defined for positive arguments.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Logarithms
Algebra
Exponents

Formulas

Multiplication Property: ln(a) + ln(b) = ln(a * b)
Division Property: ln(a) - ln(b) = ln(a / b)
Power Property: n * ln(a) = ln(a^n)

Theorems

Properties of Logarithms

Suitable Grade Level

Grades 10-12