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Question: If x, y, z are in HP, then the value of expression log<sub>e</sub> (x + z) + log<sub>e</sub> (x – 2y...

If x, y, z are in HP, then the value of expression loge (x + z) + loge (x – 2y + z) will be –

A

loge (x – z)

B

2 loge (x – z)

C

3 loge (x – z)

D

4 loge (x – z)

Answer

2 loge (x – z)

Explanation

Solution

If x, y, z are in HP, then y = 2xzx+z\frac{2xz}{x + z}

Now, loge(x + z) + loge(x – 2y + z)

= loge{(x + z) (x – 2y + z)}

=loge [(x+z)(x+z−4xzx+z)]\left\lbrack (x + z)\left( x + z - \frac{4xz}{x + z} \right) \right\rbrack [from equation. (1)]

= loge [(x + z)2 – 4xz]

= loge (x – z)2 = 2 loge(x – z).