If a=b²=c³=d^{4} then the value of _{a}(a b c d)

Correct Answer is : 1+ 12+ 13+ 141+\;1/2+\;1/3+\;1/41+21+31+41

Correct Answer is : 1+ 12+ 13+ 141+\;1/2+\;1/3+\;1/41+21+31+41

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Logarithms

Question : 48 of 85

Marks: +1, -0

a=b^{2}=c^{3}=d^{4}

\log_{a}(a b c d)

Solution:

Concept:

Express all variables in terms of a single base, then use logarithm properties.

Explanation:

Let

a = t

. Then

b^2 = t \Rightarrow b = t^{1/2}

c^3 = t \Rightarrow c = t^{1/3}

d^4 = t \Rightarrow d = t^{1/4}

Now

a b c d = t \cdot t^{1/2} \cdot t^{1/3} \cdot t^{1/4} = t^{1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}}

Therefore

\log_a (a b c d) = \log_t \left( t^{1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}} \right) = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}

Answer:

1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}

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