If x, y, z are in G.P. and a^x = b^y = c^z,th | 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 $G.P.$,we have $y^2 = xz$.Let $a^x = b^y = c^z = m$.Taking logarithm on both sides,we get $x \log a = y \log b = z \lo

Vedclass · Accepted Answer

$\log_b a = \log_c b$

Vedclass · Answer

(B) Given that $x, y, z$ are in $G.P.$,we have $y^2 = xz$.Let $a^x = b^y = c^z = m$.Taking logarithm on both sides,we get $x \log a = y \log b = z \log c = \log m$.Thus,$x = \frac{\log m}{\log a} = \log_a m$,$y = \log_b m$,and $z = \log_c m$.Since $x, y, z$ are in $G.P.$,the common ratio is constant,so $\frac{y}{x} = \frac{z}{y}$.Substituting the values of $x, y, z$ in terms of logarithms,we get $\frac{\log_b m}{\log_a m} = \frac{\log_c m}{\log_b m}$.Using the change of base formula $\log_k n = \frac{\ln n}{\ln k}$,we have $\frac{\ln m / \ln b}{\ln m / \ln a} = \frac{\ln m / \ln c}{\ln m / \ln b}$.This simplifies to $\frac{\ln a}{\ln b} = \frac{\ln b}{\ln c}$.Therefore,$\log_b a = \log_c b$.

If $x, y, z$ are in $G.P.$ and $a^x = b^y = c^z$,then

Similar Questions

If $a, b, c, d$ are positive real numbers such that $a + b + c + d = 2$,then $M = (a + b)(c + d)$ satisfies the relation

Let $S_1, S_2, \dots$ be squares such that for each $n \ge 1$,the length of a side of $S_n$ equals the length of a diagonal of $S_{n+1}$. If the length of a side of $S_1$ is $10 \text{ cm}$,then for which of the following values of $n$ is the area of $S_n$ less than $1 \text{ cm}^2$?

If the sum of $n$ terms of an $A.P.$ is $3n^2 + 5n$ and $T_m = 164$,then $m = $

If the $n^{th}$ term of a series is $3 + n(n - 1)$,then the sum of $n$ terms of the series is

If ${S_n} = nP + \frac{1}{2}n(n - 1)Q$,where ${S_n}$ denotes the sum of the first $n$ terms of an $A.P.$,then the common difference is

Vedclass Test Series

Exam Paper Generator

Online Exam Module