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)

To condense the given complex logarithm \(\log(x + 1)^2 + \log(2x - 1)^3 - \log(x)^2 - \log(2x - 1)^4 + 6\log(x + 1)\) 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:
\[
\log(x + 1)^2 + \log(2x - 1)^3 - \log(x)^2 - \log(2x - 1)^4 + 6\log(x + 1)
\]

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

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

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

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

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

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.

Learn how to condense the complex logarithm expression log(x + 1)^2 + log(2x - 1)^3 - log(x)^2 - log(2x - 1)^4 + 6log(x + 1) into a single term using logarithm properties. Understand the power, product, and quotient rules of logarithms with detailed step-by-step solutions. Suitable for high school students and anyone interested in advanced logarithmic equations.

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