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

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

#### Dr.Peterson

##### Elite Member
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]+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
2log[SUB]x[/SUB]+2log[SUB]y[/SUB](x) ......... factor out 2

= 2 * [log[SUB]x[/SUB]+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
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]+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.

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

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

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

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

#### kikithecat

##### New member
Try it this way.

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

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

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

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