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Tomorrow I have an exam about graphics and log operations. Our teacher gave us a paper with exercises to practice and one of the exercises is:

$\log_2 \sqrt[4]4$

I couldn't find the solution. My private teacher helped me but he didn't find the solution.

We don't think this log has a solution.

  • Does a solution exist for this?
  • How do you do that if there is a solution?

The exercise says the log can't be solved using a calculator.

Is my first question in Mathematics.If I am doing something wrong, please, tell me what I am doing wrong.

Peter Phipps
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spund3
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1 Answers1

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First of all, I assume that you mean the logarithm to the base $2$, i.e. that your problem is to calculate: $$\log_2(\sqrt[4]{4}).$$ Let's recall a rule for taking roots: Suppose $x>0$, and that $n$ is an integer. Then it is true that $$\sqrt[n]{x} = x^{1/n}.$$ I want to use this in your problem for the expression inside the logarithm. We get that $$\sqrt[4]{4} = 4^{1/4}.$$ Next, let's recall a rule for using logarithms. If $\log$ is any logarithm, $x>0$, and $a>0$, then $$\log(x^a) = a\log(x),$$ i.e. I can take the exponent from inside of the logarithm and pull it out. Using these two rules, I can rewrite your problem as $$\log_2(\sqrt[4]{4}) = \log_2(4^{1/4})=\frac{1}{4}\log_2(4).$$ It remains to do something about the expression $\log_2(4)$, but we can readily see what this is: $\log_2(4)$ is equal to the number $x$, such that $2^x=4$, which is just $x=2$. Thus $\log_2(4) = 2$, and so $$\log_2(\sqrt[4]{4}) = \frac{1}{4}\log_2(4) = \frac{1}{4}\cdot 2 = \frac{1}{2}.$$

Mankind
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