Test Index

Logarithms

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Question : 8 of 85

Marks: +1, -0

\;\frac{1}{\log_{xy}(xyz)}+\log_{xyz}{yz}+\;\frac{1}{\log_{zx}(xyz)}

is equal to

1
2
3
4

Solution:

Concept:

Use the property

\frac{1}{\log_a b} = \log_b a

and the rule

\log_a m + \log_a n = \log_a(mn)

.

Explanation:

Rewrite each term using the reciprocal property.

\frac{1}{\log_{xy}(xyz)} = \log_{xyz}(xy)

.

\frac{1}{\log_{zx}(xyz)} = \log_{xyz}(zx)

.

The expression becomes

\log_{xyz}(xy) + \log_{xyz}(yz) + \log_{xyz}(zx)

.

Add the logarithms:

\log_{xyz}(xy \cdot yz \cdot zx) = \log_{xyz}(x^2 y^2 z^2)

.

Since

x^2 y^2 z^2 = (xyz)^2

, we have

\log_{xyz}((xyz)^2) = 2 \cdot \log_{xyz}(xyz) = 2 \cdot 1 = 2

.

Answer:

2

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