Simplify the logarithm

Question

 

Simplify: [1/log xy(xyz)+1/log yz(xyz)+1/log zx(xyz)]

Answer 1

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!

 

Answer 2

[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