If x, y, z are in H.P.,then the value of the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given that $x, y, z$ are in $H.P.$,the middle term $y$ is the harmonic mean of $x$ and $z$,so $y = \frac{2xz}{x + z}$.We need to evaluate the expr

Vedclass · Accepted Answer

$2\log(x - z)$

Vedclass · Answer

(B) Given that $x, y, z$ are in $H.P.$,the middle term $y$ is the harmonic mean of $x$ and $z$,so $y = \frac{2xz}{x + z}$.We need to evaluate the expression $E = \log(x + z) + \log(x - 2y + z)$.Using the property $\log(a) + \log(b) = \log(ab)$,we get $E = \log[(x + z)(x - 2y + z)]$.Substitute $y = \frac{2xz}{x + z}$ into the expression:$E = \log\left[(x + z)\left(x - 2\left(\frac{2xz}{x + z}\right) + z\right)\right]$$E = \log\left[(x + z)\left(x + z - \frac{4xz}{x + z}\right)\right]$$E = \log\left[(x + z)^2 - 4xz\right]$$E = \log(x^2 + 2xz + z^2 - 4xz)$$E = \log(x^2 - 2xz + z^2) = \log(x - z)^2$Using the property $\log(a^n) = n\log(a)$,we get $E = 2\log(x - z)$.

If $x, y, z$ are in $H.P.$,then the value of the expression $\log(x + z) + \log(x - 2y + z)$ is

Similar Questions

If three geometric means are inserted between $2$ and $32$,then the third geometric mean will be

The product of three geometric means between $4$ and $\frac{1}{4}$ will be

The arithmetic mean of the nine numbers in the given set $\{9, 99, 999, \dots, 999999999\}$ is a $9$-digit number $N$,all whose digits are distinct. The number $N$ does not contain the digit

How many terms of the $A.P.$ $1, 4, 7, \ldots$ are needed to give the sum $715$?

If every term of a $G.P.$ with positive terms is the sum of its two previous terms,then the common ratio of the series is

Vedclass Test Series

Exam Paper Generator

Online Exam Module