Math Problem Statement

Condense the complex logarithm into single term log(x + 1)^2 + log(2x - 1)^3 - log(x)^2 - log(2x - 1)^4 + 6log(x + 1)

Solution

To condense the given complex logarithm log(x+1)2+log(2x1)3log(x)2log(2x1)4+6log(x+1)\log(x + 1)^2 + \log(2x - 1)^3 - \log(x)^2 - \log(2x - 1)^4 + 6\log(x + 1) into a single term, we will use the properties of logarithms. Specifically, we will use the product rule, quotient rule, and power rule of logarithms.

The expression is: log(x+1)2+log(2x1)3log(x)2log(2x1)4+6log(x+1)\log(x + 1)^2 + \log(2x - 1)^3 - \log(x)^2 - \log(2x - 1)^4 + 6\log(x + 1)

  1. Apply the power rule of logarithms to bring the exponents in front of the logarithms: 2log(x+1)+3log(2x1)2log(x)4log(2x1)+6log(x+1)2\log(x + 1) + 3\log(2x - 1) - 2\log(x) - 4\log(2x - 1) + 6\log(x + 1)

  2. Combine like terms: (2log(x+1)+6log(x+1))+(3log(2x1)4log(2x1))2log(x)(2\log(x + 1) + 6\log(x + 1)) + (3\log(2x - 1) - 4\log(2x - 1)) - 2\log(x) Simplify the coefficients: 8log(x+1)log(2x1)2log(x)8\log(x + 1) - \log(2x - 1) - 2\log(x)

  3. Use the product rule of logarithms to combine the terms into a single logarithm: log((x+1)8)log((2x1)1)log(x2)\log((x + 1)^8) - \log((2x - 1)^1) - \log(x^2)

  4. Use the quotient rule of logarithms to condense the expression further: log((x+1)8(2x1)x2)\log\left(\frac{(x + 1)^8}{(2x - 1) x^2}\right)

So, the condensed form of the given logarithmic expression is: log((x+1)8(2x1)x2)\log\left(\frac{(x + 1)^8}{(2x - 1)x^2}\right)

Would you like more details on this solution, or do you have any further questions?

Here are 5 questions you might want to ask next:

  1. How do you expand a single logarithmic term into multiple terms?
  2. What are the common properties of logarithms and how are they used in solving equations?
  3. Can you provide an example of solving a logarithmic equation step by step?
  4. How do you solve logarithmic inequalities?
  5. What are some real-world applications of logarithmic functions?

Tip: Practice combining and condensing logarithmic expressions using the product, quotient, and power rules to become more comfortable with simplifying complex logarithmic equations.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Properties of Logarithms
Logarithmic Equations

Formulas

Power rule of logarithms
Product rule of logarithms
Quotient rule of logarithms

Theorems

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Suitable Grade Level

High School