MHB How to Simplify this Complex Logarithm Expression?

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SUMMARY

The discussion focuses on simplifying the logarithmic expression $(\log_{a}(b) + \log_{b}(a) + 2)(\log_{a}(b) - \log_{ab}(b)) \cdot \log_{b}(a) - 1$. Participants emphasize the importance of showing prior attempts and understanding the properties of logarithms to facilitate better assistance. Key concepts include the change of base formula and the relationship between logarithmic identities. The conversation highlights the necessity of clarity and effort in mathematical inquiries for effective community support.

PREREQUISITES
  • Understanding of logarithmic identities
  • Familiarity with the change of base formula
  • Basic knowledge of algebraic manipulation
  • Experience with $\LaTeX$ typesetting for mathematical expressions
NEXT STEPS
  • Research the properties of logarithms, including product and quotient rules
  • Learn how to apply the change of base formula in logarithmic expressions
  • Practice simplifying complex logarithmic expressions with examples
  • Explore advanced topics in logarithmic functions, such as their applications in calculus
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Students, educators, and anyone seeking to deepen their understanding of logarithmic expressions and improve their problem-solving skills in mathematics.

Elena1
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1) $( \log_{a}\left({b}\right)+\log_{b}\left({a}\right)+2) (\log_{a}\left({b}\right)-\log_{ab}\left({b)}\right)* \log_{b}\left({a}\right)-1=$?
 
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Elena said:
1) ( \log_{a}\left({b}\right)+\log_{b}\left({a}\right)+2) (\log_{a}\left({b}\right)-\log_{ab}\left({b)}\right)* \log_{b}\left({a}\right)-1=?

You need to enclose your $\LaTeX$ code within tags, such as $$$$.

We also expect for people posting questions to show what they have tried, where they are stuck, so we can offer better, more specific help. This is outlined in MHB rule #11:

MHB Rules said:
Show some effort. If you want help with a question we expect you to show some effort. Effort might include showing your work, learning how to typeset equations using $\LaTeX$, making your question clearer, titling threads effectively and posting in the appropriate subforum, making a genuine attempt to understand the given help before asking for more help, and learning from previously asked questions. Moderators reserve the right to close threads in cases where the member is not making a genuine effort (particularly if the member is flooding the forums with multiple questions of the same type). You also should remember that all contributors to MHB are unpaid volunteers and are under no obligation to answer a question.
 

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