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anonymous

  • one year ago

Please help!!!! 3log(x^2) + 4log(2x)=2

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  1. mathstudent55
    • one year ago
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    CAn you use the rules of logs to come up with a single log expression on the left side?

  2. anonymous
    • one year ago
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    I know the first couple of steps...

  3. anonymous
    • one year ago
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    Not sure what you're asking? Sorry.

  4. mathstudent55
    • one year ago
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    Can you show the steps you already did?

  5. anonymous
    • one year ago
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    3log(x^2) + 4log(2x)=2 log(x^2)^3 + log(2x)^4=2 log (x^6) + log (16x^4)=2

  6. anonymous
    • one year ago
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    log (x^24) = 2?

  7. mathstudent55
    • one year ago
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    The last step is not correct, but until the previous step you are correct.

  8. anonymous
    • one year ago
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    log (x^10) = 2?

  9. mathstudent55
    • one year ago
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    \(3 \log x^2 + 4 \log 2x = 2\) \(\log (x^2)^3 + \log(2x)^4 = 2\) \(\log x^6 + \log (2^4 x^4) = 2\)

  10. mathstudent55
    • one year ago
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    Now we use the rule of logs: \(\log a + \log b = \log (ab) \)

  11. anonymous
    • one year ago
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    So, log (16x^24) = 2?

  12. mathstudent55
    • one year ago
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    \(\log (x^6 \cdot 2^4x^4) = 2\) \(\log (2^4 x^{10}) = 2\)

  13. anonymous
    • one year ago
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    oh you add 6 and 4..

  14. mathstudent55
    • one year ago
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    Remember, when you multiply powers with the same base, you ADD the exponents.

  15. mathstudent55
    • one year ago
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    The rule is: \(a^m \cdot a^n = a^{m + n} \)

  16. anonymous
    • one year ago
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    So for the next step... log(2)= 16x^10?

  17. anonymous
    • one year ago
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    Right, okay.

  18. mathstudent55
    • one year ago
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    Ok. Now we use the definition of log. Is this log base 10 or natural log?

  19. anonymous
    • one year ago
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    log base 10

  20. mathstudent55
    • one year ago
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    Definition of log: \(\large \log_b x = y \iff b^y = x\)

  21. mathstudent55
    • one year ago
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    Now let's use the last equation we have as the left side of the definition of log, and change it into the exponential version on the right side.

  22. mathstudent55
    • one year ago
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    \(\log (2^4 x^{10}) = 2~~~\iff~~~10^2 = 2^4x^{10}\) Do you understand this step?

  23. anonymous
    • one year ago
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    Ah, yes

  24. mathstudent55
    • one year ago
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    Great. Now let's continue solving for x.

  25. anonymous
    • one year ago
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    100=16x^10 Divide 16 6.25=x^10 6.25^(1/10) x=1.2011

  26. mathstudent55
    • one year ago
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    \(2^4x^{10} = 10^2\) \(x^{10} = \dfrac{10^2}{2^4} \) \(x^{10} = \left( \dfrac{10}{2^2} \right)^2\) \(x^{10} = \left( \dfrac{5}{2} \right)^2\) \(\left (x^{10} \right)^{\frac{1}{10}} = \left( \left( \dfrac{5}{ 2} \right)^2\ \right)^{\frac{1}{10}} \)

  27. anonymous
    • one year ago
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    I understand now, thanks so much!

  28. mathstudent55
    • one year ago
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    \(x = \left( \dfrac{5}{2} \right) ^ {\frac{1}{5}} \) \(x = \sqrt[5] {\dfrac{5}{2} } \)

  29. mathstudent55
    • one year ago
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    You're welcome.

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