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Math Help - Compare numbers with logarithm exponents

  1. #1
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    Compare numbers with logarithm exponents

    Can you please compare the numbers 3^log4 of 5 and 5^ log4 of 3, just to make it clear the bases are equal and are 4. Please help me with this problem
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  2. #2
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    Re: Compare numbers with logarithm exponents

    Quote Originally Posted by Jkanariya View Post
    Can you please compare the numbers 3^log4 of 5 and 5^ log4 of 3, just to make it clear the bases are equal and are 4. Please help me with this problem
    I used a circuitous method involving taking the log of both sides and then using the change of base rule.

    3^{\log_4(5)} = 5^{\log_4(3)}

    \log_4(5)\ln(3) = \log_4(3)\ln(5)

    \dfrac{\ln(5)\ln(3)}{\ln(4)} = \dfrac{\ln(3)\ln(5)}{\ln(4)}

    Hence they're equal.
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  3. #3
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    Re: Compare numbers with logarithm exponents

    Hello, Jkanariya!

    \text{Compare }\:3^{\log_45}\,\text{ and }\,5^{\log_43}

    They are equal!


    \text{Let }\,3^{\log_45} \,=\,P

    \text{Take logs, base 4:}
    . . \log_4\left(3^{\log_45}\right) \:=\:\log_4P \quad\Rightarrow\quad (\log_45)(\log_43) \:=\:\log_4P \;\;{\bf[1]}


    \text{Let }\,5^{\log_43} \,=\,Q

    \text{Take logs, base 4:}
    . . \log_4\left(5^{\log_43}\right) \:=\:\log_4Q \quad\Rightarrow\quad (\log_43)(\log_45) \:=\:\log_4Q\;\;{\bf[2]}


    \text{Equate }{\bf[1]}\text{ and }{\bf[2]}\!:\;\;\log_4P \:=\:\log_4Q \quad\Rightarrow\quad P \:=\:Q


    \text{Therefore: }\:3^{\log_45} \;=\;5^{\log_43}


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  4. #4
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    Re: Compare numbers with logarithm exponents

    Thanks but there is something i dont understand why is Log4 of P = (log4 of 3)(log4 of 5)?
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