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Thread: Solving Without a Calculator?

  1. #1
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    Solving Without a Calculator?

    Evaluate:

    log4 (1/8)

    how can i solve without a calculator. i know this answer is -1.5

    = log(1/8)
    log 4
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  2. #2
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    I would not use a change of base rule here. What you want is $\displaystyle \frac{1}{8}$ in terms of 4's, i.e., $\displaystyle \frac{1}{2}\cdot \frac{1}{4} = \left(\frac{1}{4}\right)^{0.5}\cdot \frac{1}{4}$. Now you can use your log rules to calculate the actual value.
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  3. #3
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    Hello, danielh9103!

    Evaluate: .$\displaystyle \log_4\!\left(\frac{1}{8}\right)$

    Let: .$\displaystyle \log_4\!\left(\frac{1}{8}\right) \:=\:P$

    Then: .$\displaystyle 4^P \:=\:\dfrac{1}{8} \quad\Rightarrow\quad (2^2)^P \:=\:\dfrac{1}{2^3} \quad\Rightarrow\quad 2^{2P} \:=\:2^{-3}$

    Hence: .$\displaystyle 2P \:=\:-3 \quad\Rightarrow\quad P \:=\:-\frac{3}{2}$


    Therefore: .$\displaystyle \log_4(\frac{1}{8}) \;=\;-\frac{3}{2}$

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  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
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    log(1/8)=log(1/2^3)=log(2^(-3))=-3log2=-3/2
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