# Logarithm equation

#### roger

hi

could anyone tell me where I went wrong ?

simultaneously solve

2logbase2 y = logbase4 3 + logbase2 x

3^y = 9^x

But for the top I get y = 3 root x

and bottom I get y=3x

so what's gone wrong ?

thanks

roger

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#### dextercioby

Homework Helper
Use $\ln$

The second becomes

$$y\ln 3=2x\ln 3\Rightarrow y=2x$$ (1)

while the first

$$2\ln y=\frac{\ln 3}{2}+\ln x$$ (2)

Solve the simple system (1) + (2)

Daniel.

#### roger

do you mean log base e when you state ln ?

But my working out is as follows and please tell me what went wrong :

logbase2 y ^2 = logbase2 root3 + logbase2 x

= log base 2 xroot3 = log base2 y ^2
then remove logs and square both sides to get y = 3 root x

#### HallsofIvy

Homework Helper
Sorry, but I can't find anything that's gone right.

9= 32 so 9x= 32x.

The second equation is 3y= 32x which gives y= 2x, not y= 3x.

The first equation is 2 log2(y)= log4(3)+ log2(x)
log2(y2)- log2(x)= log4(3)
log2(y2/x)= log4(3)

If z= log4(3) then 3= 4z= (22)z= 22z. Now taking log2 of both sides, gives log2(3)= 2z so log4(3)= (1/2)log2(3)= log2(31/2).

Putting those together, log2(y2/x)= log2(31/2) so that y2/x= 31/2 or y= 31/4x1/2, not y= 3 x1/2.

That is, we have 2x= 31/4x1/2.

One obvious answer to that is x= 0.

Squaring both sides, 3x2= 31/2x and if x is not 0,
4x= 31/2 so x= 31/2/4 is another solution.

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#### dextercioby

Homework Helper
Halls,there's no such thing as logarithm from 0...That "x" equal 0 doesn't satisfy the first equation.

Daniel.

#### roger

HallsofIvy said:
Sorry, but I can't find anything that's gone right.

9= 32 so 9x= 32x.

The second equation is 3y= 32x which gives y= 2x, not y= 3x.

The first equation is 2 log2(y)= log4(3)+ log2(x)
log2(y2)- log2(x)= log4(3)
log2(y2/x)= log4(3)

If z= log4(3) then 3= 4z= (22)z= 22z. Now taking log2 of both sides, gives log2(3)= 2z so log4(3)= (1/2)log2(3)= log2(31/2).

Putting those together, log2(y2/x)= log2(31/2) so that y2/x= 31/2 or y= 31/4x1/2, not y= 3 x1/2.

That is, we have 2x= 31/4x1/2.

One obvious answer to that is x= 0.

Squaring both sides, 3x2= 31/2x and if x is not 0,
4x= 31/2 so x= 31/2/4 is another solution.
why ?

and dextercioby, x cant be negative but cant it tend to zero to give the power of negative infinity ?

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#### dextercioby

Homework Helper
No,no.That is x=0...It's an exact solution.It's not acceptable,as 0 is not in the domain of logarithm (in any base).

Daniel.

#### HallsofIvy

$$x= \frac{\sqrt{3}}{4}, y= \frac{\sqrt{3}}{2}$$ is the only solution.