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)]

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[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

Shaina T. | Harvard Graduate and Experienced TutorHarvard Graduate and Experienced Tutor

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

You also need to know the * product rule of logarithms*: the log of a product is the sum of the logs. In short, log

**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!**

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