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What logarithm rule can convert:

$$\left(\frac n4\right)^i = 1$$

to:

$$i = log_4(n)$$

When I view cheat-sheets for logarithm rules, I only see conversions where both sides of the equation have log in it. Thank you.

Stefan4024
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user84756
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  • "I only see conversions where both sides of the equation have log in it.": are you sure ? –  May 03 '20 at 13:15
  • I randomly took a few cheat sheets with logarithms. All gave the solution. –  May 03 '20 at 13:17

2 Answers2

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The basic rule

$$\log_ax^n=n\log_ax$$

so

$$\left(\frac n4\right)^i=1\iff i\log\frac n4=\log 1=0$$

What you have seems to be incorrect, thus.

DonAntonio
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No rule can get you from the first equation to the second. What is true is that $$\begin{align} 0=\log_41&=\log_4\left(\left(\frac{n}{4}\right)^i\right)\\ &=i\log_4\left(\frac{n}{4}\right)\\ &=i(\log_4n-\log_44)\\ &=i(\log_4n-1) \end{align}$$ and this is only satisfied when $i=0$ or $n=4$ (which makes perfect sense by looking at your first equation).

Rick Decker
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