Logarithms Question Help

[modern hinge]

The first to eq are given data need to find the 3rd eq value.... ans is 4.25 but i need solving steps

[full thistle]

first of all, do you know the identity log_a(b) = log(b)/log(a) ?

[modern hinge]

loga-logb=loga/logb right?

[full thistle]

no, that's not right

$\log_a(b) = \frac{\log (b)}{\log (a)}$, and $\log (a) - \log(b) = \log (\frac{a}{b})$

[tough quartz]

HChan

[full thistle]

don't get them mixed up

[modern hinge]

Change base is the first one

[full thistle]

ye

s

[modern hinge]

Yeah i know it

[full thistle]

okay, so then

$\log_x y + \log_y x = \frac{\log y}{\log x} + \frac{\log x}{\log y}$

[tough quartz]

HChan

[full thistle]

so that's the first thing

for sake of simplicity, I'll call log y Y and log x X

[modern hinge]

ok

[modern hinge]

i did not look at it this way

[full thistle]

now I can rewrite the question as this:

Given $\begin{cases}
X + Y = \frac{5}{2} \
XY = 1
\end{cases}$,
what is $\frac{X}{Y} + \frac{Y}{X}$?

[tough quartz]

HChan

[full thistle]

now can you solve this

[modern hinge]

Yes common denominator then you will x^2+y^2/xy

then xy is 1

and x^2+y^2 = (x+y)^2-2xy

[full thistle]

there you go