I'm unable to solve this equation further. Could someone have a shot at it and try to solve it and explain it to me please? The equation is $$2\log_{2}(\log_{2}(x))-\log_{2}(\log_{2}2\cdot 2^{0.5}x)=1.$$ I am trying to learn the general rule for solving this kind of log problems. So it does involve a concept and kindly donut flag it. Two people have already given me some good answers.
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Is that actually $ \ (2.2)^{0.5} \ x \ $ , or is it supposed to be $ \ 2 \cdot 2^{0.5} \ x \ $ ? – colormegone Jun 28 '16 at 03:14
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1Could you place parentheses around appropriate parts so as to prevent confusion? – Drew Christensen Jun 28 '16 at 03:17
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It is $2⋅2^{0.5}.x$ – Ujjwal Jun 28 '16 at 03:20
2 Answers
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1) Perform algebraic rearrangement
$$2\log_{2}(\log_{2}(x))=1+\log_{2}(\log_{2}(2\cdot 2^{0.5}x))$$
2) Apply laws of logarithms
$$\log_2((\log_2(x))^2)=\log_{2}(2\log_{2}(2\cdot 2^{0.5}x))$$
3) Raise 2 to the power of both sides
$$(\log_2(x))^2=2\log_{2}(2\cdot 2^{0.5}x)$$
4) Apply the laws of logarithms once more
$$(\log_2(x))^2=2\log_2(2\cdot2^{0.5})+2\log_2(x)=3+2\log_2(x)$$
Notice that this can be arranged into a quadratic in $\log_2(x)$, namely
$$(\log_2(x))^2-2\log_2(x)-3=0$$
which can be factored as
$$(\log_2(x)+1)(\log_2(x)-3)=0$$
Therefore, $\log_2(x)=-1$ or $\log_2(x)=3$. This is equivalent to $x=1/2$ or $x=8$. But we aren't yet done! Notice that for 1/2 to be a solution, logarithms have to be defined for negative numbers. This cannot be. Thus, 8 is the only solution.
Nathaniel B
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Put $\log_2(x)=X$. You have $$\log_2\frac{X^2}{3/2+X}=1=\log_2(2)$$ $$X^2=3+2X\iff X=3\text{ or -1}\iff \log_2 (x)=3\text{ or } \log_2(x)=-1$$ You have to discard $\log_2(x)=-1$ because of the proposed equation.Hence $$\color{red}{x=8}$$
Piquito
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