Simplify: [1/log xy(xyz)+1/log yz(xyz)+1/log zx(xyz)]
Simplify: [1/log xy(xyz)+1/log yz(xyz)+1/log zx(xyz)]
[1/log xy(xyz)+1/log yz(xyz)+1/log zx(xyz)]
= log xy/log xyz + log yz/log xyz + log zx/log xyz, using the change of base formula
= log (xy*yz*zx)/log xyz, addtion property of log function
= log (xyz)^2 /log xyz
= 2log xyz/log xyz, power property of log function
= 2
To simplify this expression, you need to know some key properties of logarithms:
You need to know the change of base formula, which is a very important property of logarithms. In short, log_{a}(x) = (log_{b}(x)/log_{b}(a)); any value of b will work, but it is generally easiest to use base 10.
You also need to know the product rule of logarithms: the log of a product is the sum of the logs. In short, log_{a}(xy) = log_{a}(x) + log_{a}(y)
Step 1, write the problem: 1/log_{xy}(xyz) + 1/log_{yz}(xyz) + 1/log_{zx}(xyz)
Step 2, apply change of base formula, using base 10, to each separate logarithm in the expression:
log_{xy}(xyz) = log (xyz) / log (xy)
log_{yz}(xyz) = log (xyz) / log (yz)
log_{zx}(xyz) = log (xyz) / log (zx)
Step 3, replace the original logarithms with the results of your change of base formula:
1/[log (xyz) / log (xy)] + 1/[log (xyz) / log (yz)] + 1/[log (xyz) / log (zx)]
Step 4, rewrite each quotient (to divide by a fraction, you multiply by the inverse of the fraction)
log (xy)/log (xyz) + log (yz)/log (xyz) + log (zx)/log (xyz)
Step 5, combine all three fractions, since they share a common denominator:
[log (xy) + log (yz) + log (zx)] / log (xyz)
Step 6, apply product rule log_{a}(xy) = log_{a}(x) + log_{a}(y)
[log x + log y + log y + log z + log z + log x]/[log x + log y + log z]
Step 7, simplify the numerator of the expression:
(2 log x + 2 log y + 2 log z)/(log x + log y + log z)
Step 8, factor out 2 from numerator:
2(log x + log y + log z)/(log x + log y + log z)
Step 9, cancel out (log x + log y + log z) from both the numerator and the denominator, leaving 2 as your answer.
The simplified answer is 2!