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anonymous

  • one year ago

I've been having a touch of trouble with solving logarithms with uneven bases. For example 8^x-10=3^x+2 Now I know that in order to solve I need to log8^x-10=log3^x+2 then you move the power out in front of the logs like so x-10 log8=x+2 log3 this is where I stumble, you find the approximate value of the log soo (x-10)(.903)=(x-2)(4.77) It's after this step were I get lost.

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  1. phi
    • one year ago
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    if it was (x-10)*2 = (x-2)*3 what would you do? I would distribute, and combine "like terms"

  2. phi
    • one year ago
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    btw, I assume you meant log(3) = 0.47712 (not 4.77)

  3. phi
    • one year ago
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    so do this \[ (x-10)(.903)=(x-2)(0.477) \\ 0.903x -9.03 = 0.477x -0.954 \]

  4. phi
    • one year ago
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    can you finish?

  5. anonymous
    • one year ago
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    Yea one sec

  6. anonymous
    • one year ago
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    .426x=-.051 -/051/.426=-.119

  7. anonymous
    • one year ago
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    Except x is supposed to = 23.44 ... So I did something wrong

  8. phi
    • one year ago
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    ok, then step by step \[ 8^{(x-10)}=3^{(x+2)} \\ \log\left( 8^{(x-10)}\right) = \log\left( 3^{(x+2)} \right) \] \[ (x-10) \log(8) = (x+2) \log(3) \] rather than approximate the logs, treat them as numbers (which is what they are) \[ x \log(8) - 10\log(8) = x \log(3) + 2 \log(3) \\ x \log(8) - x \log(3) = 10\log(8) + 2 \log(3) \\ x ( \log(8) - \log(3) ) = 10\log(8) + 2 \log(3) \\ \] and finally \[ x = \frac{10\log(8) + 2 \log(3) }{ \log(8) - \log(3)} \]

  9. phi
    • one year ago
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    btw, my first equation I managed to write (x-2) instead of x+2. That was bad. but, do you follow the steps up above? to get the final answer you need a calculator

  10. anonymous
    • one year ago
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    I get 10.57.... not 23.44

  11. phi
    • one year ago
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    what do you get for the top? (btw, I'm getting 23.44102, so it does work out)

  12. anonymous
    • one year ago
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    I type in 10log(8)+2log(3) and get 9.985...

  13. phi
    • one year ago
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    ok, now the bottom

  14. anonymous
    • one year ago
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    .4259...

  15. phi
    • one year ago
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    I get 0.425969 which rounds to 0.4260 to 4 digits but ok now 9.985/ 0.426

  16. phi
    • one year ago
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    if you keep more digits , you will get a more accurate answer

  17. phi
    • one year ago
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    if you type the expression into google like this (10 log 8 + 2 log 3)/(log 8 - log 3)= you get out 23.441021894

  18. phi
    • one year ago
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    but I would learn how to use a calculator (it is a useful to know how to do complicated expressions)

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