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Math Help - Logarithms

  1. #1
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    Logarithms

    Please help me with the following problems, as I am unsure how to solve them:

    1.) log(x+1)^0.25 = 0.50

    2.) 2^(x-1) = 3^(x-1)

    3.) 600 = 1000(.04^(.3^x))

    Your assistance would be much appreciated.
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  2. #2
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    Hello, currypuff!

    1)\;\;\log(x+1)^\frac{1}{4} \:= \:\frac{1}{2}
    I will assume that this log is base-ten.

    We have: . \frac{1}{4}\cdot\log(x+1) \:=\:\frac{1}{2} \quad\Rightarrow\quad \log(x+1) \:=\:2

    Then: . x + 1 \:=\:10^2\quad\Rightarrow\quad \boxed{x \:=\:99}




    2)\;\;2^{x-1} \:= \:3^{x-1}

    Take logs: . \ln\left(2^{x-1}\right) \:=\:\ln\left(3^{x-1}\right) \quad\Rightarrow\quad (x-2)\ln(2) \;=\;(x-1)\ln(3)

    . . x\ln(2) - \ln(2) \:=\:x\ln(3) - \ln(3) \quad\Rightarrow\quad x\ln(2) - x\ln(3) \:=\:\ln(2) - \ln(3)

    Factor: . x\left[\ln(2)-\ln(3)\right] \;=\;\ln(2)-\;\ln(3)

    Therefdore: . x \;=\;\frac{\ln(2)-\ln(3)}{\ln(2)-\ln(3)}\quad\Rightarrow\quad\boxed{x \:=\:1}




    3)\;\;1000\left(0.04^{0.3^x}\right) \:=\:600

    Divide by 1000: . 0.04^{0.3^x} \:=\:0.6

    Take logs: . \ln\left(0.04^{0.3^x}\right) \:=\:\ln(0.6) \quad\Rightarrow\quad 0.3^x\cdot\ln(0.04) \:=\:\ln(0.6)

    . . Then we have: . 0.3^x \;=\;\frac{\ln(0.6)}{\ln(0.04)}


    Take logs: . \ln\left(0.3^x\right) \;=\;\ln\left[\frac{\ln(0.6)}{\ln(0.04)}\right] \quad\Rightarrow\quad x\ln(0.3) \;=\;\ln\left[\frac{\ln(0.6)}{\ln(0.04)}\right]


    \text{Therefore: }\;x \;=\;\frac{\ln\left[\dfrac{\ln(0.06)}{\ln(0.04)}\right]}{\ln(0.3)}

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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by currypuff View Post
    Please help me with the following problems, as I am unsure how to solve them:

    1.) log(x+1)^0.25 = 0.50
    use the rules:

    (1) \log_a x^n = n \log_a x

    and

    (2) \log_a b = c \Longleftrightarrow a^c = b

    identify what are your a,b, and c and then just follow the formula

    2.) 2^(x-1) = 3^(x-1)
    log both sides.

    you get \ln 2^{x - 1} = \ln 3^{x - 1}

    then use law (1) from above and solve for x. remember, ln(2) and ln(3) are constants

    3.) 600 = 1000(.04^(.3^x))
    divide through by 1000

    you get 0.6 = 0.04^{0.3^x}

    log both sides, you get:

    \ln 0.6 = \ln 0.04^{0.3^x}

    apply law (1) above:

    \ln 0.6 = 0.3^x \ln 0.04

    divide by \ln 0.04, you get:

    \frac {\ln 0.06}{\ln 0.04} = 0.3^x

    log both sides again:

    \ln \left( \frac {\ln 0.6}{\ln 0.04} \right) = \ln 0.3^x

    now apply law (1) again and continue
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