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Alg II / Pre-Calc w/ Logs: Given log_x(y)+log_y(x)=3, find log_x(y^2)+log_y(x^2)

kikithecat

New member
Joined
Nov 24, 2017
Messages
5
Hi there.

The question I have to complete is this : "Let x and y be two positive real numbers such that log[SUB]x[/SUB]y+log[SUB]y[/SUB]x = 3. Find the value of log[SUB]x[/SUB]y[SUP]2[/SUP]+log[SUB]y[/SUB]x[SUP]2[/SUP]." Now, I have simplified the equation down to 2log[SUB]x[/SUB](y)+2log[SUB]y[/SUB](x), but I don't know how to find the value after that. Would the 3 from the original equation have to be squared as well? Any guidance from there would be helpful.

Thanks!
 

Subhotosh Khan

Super Moderator
Staff member
Joined
Jun 18, 2007
Messages
18,086
Hi there.

The question I have to complete is this : "Let x and y be two positive real numbers such that log[SUB]x[/SUB]y+log[SUB]y[/SUB]x = 3. Find the value of log[SUB]x[/SUB]y[SUP]2[/SUP]+log[SUB]y[/SUB]x[SUP]2[/SUP]." Now, I have simplified the equation down to 2log[SUB]x[/SUB](y)+2log[SUB]y[/SUB](x), but I don't know how to find the value after that. Would the 3 from the original equation have to be squared as well? Any guidance from there would be helpful.

Thanks!
2log[SUB]x[/SUB](y)+2log[SUB]y[/SUB](x) ......... factor out 2

= 2 * [log[SUB]x[/SUB](y)+log[SUB]y[/SUB](x)] ...... continue....
 

Dr.Peterson

Elite Member
Joined
Nov 12, 2017
Messages
2,745
Hi there.

The question I have to complete is this : "Let x and y be two positive real numbers such that log[SUB]x[/SUB]y+log[SUB]y[/SUB]x = 3. Find the value of log[SUB]x[/SUB]y[SUP]2[/SUP]+log[SUB]y[/SUB]x[SUP]2[/SUP]." Now, I have simplified the equation down to 2log[SUB]x[/SUB](y)+2log[SUB]y[/SUB](x), but I don't know how to find the value after that. Would the 3 from the original equation have to be squared as well? Any guidance from there would be helpful.

Thanks!
Don't square; just factor out the 2!
 

kikithecat

New member
Joined
Nov 24, 2017
Messages
5
2log[SUB]x[/SUB](y)+2log[SUB]y[/SUB](x) ......... factor out 2

= 2 * [log[SUB]x[/SUB](y)+log[SUB]y[/SUB](x)] ...... continue....
How would the 3 in the original equation be integrated into this? Would it be squared, would it be left as 3...?
 

JeffM

Elite Member
Joined
Sep 14, 2012
Messages
3,174
Hi there.

The question I have to complete is this : "Let x and y be two positive real numbers such that log[SUB]x[/SUB]y+log[SUB]y[/SUB]x = 3. Find the value of log[SUB]x[/SUB]y[SUP]2[/SUP]+log[SUB]y[/SUB]x[SUP]2[/SUP]." Now, I have simplified the equation down to 2log[SUB]x[/SUB](y)+2log[SUB]y[/SUB](x), but I don't know how to find the value after that. Would the 3 from the original equation have to be squared as well? Any guidance from there would be helpful.

Thanks!
Try it this way.

\(\displaystyle \text {GIVEN: } x,\ y \in \mathbb R^+ \text { and } log_x(y) + log_x(y) = 3.\)

\(\displaystyle \text {Let } u = log_x(y) \text { and } v = log_y(x) \implies\)

\(\displaystyle u + v = 3 \text { and } log_x(y^2) = 2u \text { and } log_y(x^2) = 2v \implies \)

\(\displaystyle log_x(y^2) + log_y(x^2) = 2u + 2v = 2(u + v) = WHAT?\)
 

kikithecat

New member
Joined
Nov 24, 2017
Messages
5
Try it this way.

\(\displaystyle \text {GIVEN: } x,\ y \in \mathbb R^+ \text { and } log_x(y) + log_x(y) = 3.\)

\(\displaystyle \text {Let } u = log_x(y) \text { and } v = log_y(x) \implies\)

\(\displaystyle u + v = 3 \text { and } log_x(y^2) = 2u \text { and } log_y(x^2) = 2v \implies \)

\(\displaystyle log_x(y^2) + log_y(x^2) = 2u + 2v = 2(u + v) = WHAT?\)
Oh, I see now! Using a simpler version of the problem was helpful. I was over-analyzing it, believing that I needed to find the values of x and y when it was really as simple as that. Thank you!
 
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