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Thread: Log Help!!

  1. #1
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    Log Help!!

    Hi guys I have a few questions that I am totally lost on.

    1) Write as a single loagrithm: 4lnx - 2(lnx^3 + 4lnx)

    2) Solve: 2^x = 3^(x+3)

    3) Solve: log(x^2 -1) = 2 + log (x+1)

    Thanks in advance!!
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  2. #2
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    Quote Originally Posted by antz215 View Post
    Hi guys I have a few questions that I am totally lost on.

    1) Write as a single loagrithm: 4lnx - 2(lnx^3 + 4lnx)
    what have you tried?

    here are the rules you need:

    $\displaystyle \log_a(x^n) = n \log_a x$

    $\displaystyle \log_a(xy) = \log_ax + \log_a y$

    $\displaystyle \log_a \left( \frac xy \right) = \log_a x - \log_a y$


    2) Solve: 2^x = 3^(x+3)
    take the log of both sides. use the laws of logs i gave you above to simplify the expression.

    3) Solve: log(x^2 -1) = 2 + log (x+1)
    same idea here. you may also want to use: $\displaystyle \log_a b = c \Longleftrightarrow a^c = b$

    try them and see what you get
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  3. #3
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    Yeah, I have all those rules, but I don't completely understand them. I'm awful at logs. Thanks anyway!
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by antz215 View Post
    Hi guys I have a few questions that I am totally lost on.

    1) Write as a single loagrithm: 4lnx - 2(lnx^3 + 4lnx)
    $\displaystyle 4 \ln x - 2( \ln (x^3) + 4 \ln x) = \ln (x^4) - 2( \ln (x^3) + \ln (x^4))$ ..................since $\displaystyle n \log_a x = \log_a (x^n)$

    $\displaystyle = \ln (x^4) - 2(\ln (x^7))$ ...................since $\displaystyle \log_a x + \log_a y = \log_a xy$

    $\displaystyle = \ln (x^4) - \ln (x^{14})$ .....................since $\displaystyle n \log_a x = \log_a (x^n)$

    $\displaystyle = \ln \left( \frac {x^4}{x^{14}}\right)$ .............................since $\displaystyle \log_a x - \log_a y = \log_a \frac xy$

    $\displaystyle = \ln (x^{-10})$

    2) Solve: 2^x = 3^(x+3)
    $\displaystyle 2^x = 3^{x + 3}$ ........................log both sides

    $\displaystyle \Rightarrow \ln 2^x = \ln 3^{x + 3}$ ......................use the power rule for logs

    $\displaystyle \Rightarrow x \ln 2 = (x + 3) \ln 3$ .................distribute

    $\displaystyle \Rightarrow x \ln 2 = x \ln 3 + 3 \ln 3$ ...............get all x's to one side

    $\displaystyle \Rightarrow x \ln 2 - x \ln 3 = \ln 27$ .................factor out the x

    $\displaystyle \Rightarrow x (\ln 2 - \ln 3) = \ln 27$ .................solve for x

    $\displaystyle \Rightarrow x = \frac {\ln 27}{\ln 2 - \ln 3}$ .....................simplify

    $\displaystyle \Rightarrow x = \frac {\ln 27}{\ln \frac 23}$

    3) Solve: log(x^2 -1) = 2 + log (x+1)
    i leave this one to you. try to do the similar manipulations as i did in the previous questions
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  5. #5
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    thanks a lot! i'll try my best with the last one!
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  6. #6
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    i tried subtracting log (x+1) to the other side, and got:

    log( (x+1)(x-1)/x+1) = 2

    then i canceled the (x+1)'s

    So right now i have log (x-1) = 2

    am I at least going in the right direction? haha
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  7. #7
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