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Question

Find all the real solutions to the logarithmic equation
ln(x4)+ln(x2)−ln(x3)−2=7

A
91/3
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B
e3
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C
1
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D
no real solutions
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E
0
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Solution

The correct option is B e3
Let both sides be exponents of the base e. The equation ln(x4)+ln(x2)ln(x3)2=7 that is ln(x4)+ln(x2)ln(x3)=9can be rewritten as eln(x4)+ln(x2)ln(x3)=e9 or eln(x4x2/(x3))=e9 that is eln(x3)=e9.

By now you should know that when the base of the exponent and the base of the logarithm are the same, the left side can be written x. The equation eln(x3)=e2 can now be written as x3=e9 that is

x=(e9)1/3
x=e3

Therefore, x=e3.

Hence option B is correct.

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