Logs, 2s, and 3s

Algebra Level 2

Find all real solutions $x$ to

$3\log_2(x) - 1 = \log_2\left(\frac32 x-1\right).$

Enter your answer as the sum of all such $x$ .

1 -1 2 0

1 solution

Rishabh Jain
Feb 16, 2016

Equation can be rearranged to: $\Large \log_2(\dfrac{x^3}{2}) = \log_2(\dfrac{3x}{2} -1)$ $\Large \Rightarrow \dfrac{x^3}{2}=\dfrac{3x}{2}-1$ $\Large \Rightarrow x^3-3x+2$ $\Large (x-1)^2(x+2)=0$ $\Large \Rightarrow x=1,-2$ But since domain of a log function is set of positive reals we can easily reject $x=-2$ . Hence the only solution which hence is also the sum of solutions of the equation is $\huge\boxed1$

I actually forgot to take note that it must be $x>\frac { 2 }{ 3 }$ in this case.

Kenneth Choo - 5 years, 3 months ago

i misread 3x/2-1 as3x/2+1.

Sachetan Debray - 2 years, 4 months ago

How come you can throw out the (- 1) on the left-hand side of the equation, when it is rearranged?

A Former Brilliant Member - 2 years, 2 months ago

I didn't throw it out. Look its accommodated as 2 in $\frac{x^3}2$

Rishabh Jain - 2 years ago

I solved it but took me 10 minutes total because solving the cubic equation without calculator.

Elvarn Eng Hui - 1 year, 12 months ago

Isn't this problem a bit overrated?

Mehul Arora - 5 years, 3 months ago

Absolutely not... People always face trouble when they have to reject obtained solutions ( And above that this problem is MCQ which make it more dangerous for people who take log casually ;-) See not even 50% of the solvers got this problem right.. :-). Might be after seeing it a Level 4 remaining solvers will solve this problem cautiously and the problem's level surely gonna drop... Sorry for writing too long ;-}

Rishabh Jain - 5 years, 3 months ago

Mayn

As they say, Tu toh senti ho gaya :P

Anyway, well, maybe the problem is not that overrated :P

Mehul Arora - 5 years, 3 months ago

@Mehul Arora – Yeah .. Typical Hindi .... :-)

Rishabh Jain - 5 years, 3 months ago

Logs, 2s, and 3s

1 solution

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