First slide

Geometric progression

Question

If $x, y, z$ ,are in GP and $a^{x} = b^{y} = c^{z}$ , then

Moderate

A

$\log_{a} c = \log_{b} a$
B

$\log_{b} a = \log_{c} b$
C

$\log_{c} b = \log_{a} c$
D

None of these

Solution

Given, x, y, z are in GP and $a^{x} = b^{y} = c^{z}$ Since, x, y, z are in GP.
$\Rightarrow y^{2} = xz . . . . . . (i)$
we have
$a^{x} = b^{y} = c^{z} = k (say); k > 0$
On taking log both sides, we get
$\begin{aligned} xlog a = ylog b = zlog c = \log k \\ \Rightarrow x = \frac{\log k}{\log a}, y = \frac{\log k}{\log b}, z = \frac{\log k}{\log c} \end{aligned}$
On substituting the values of x,y and z in Eq. (i), we get
${(\frac{\log k}{\log b})}^{2} = \frac{\log k}{\log a} \cdot \frac{\log k}{\log c}$
$\begin{array}{ll} \Rightarrow \frac{(\log k)^{2}}{(\log b)^{2}} = \frac{(\log k)^{2}}{\log a \cdot \log c} \\ \Rightarrow (\log b)^{2} = \log a \cdot \log c \\ \Rightarrow \frac{\log a}{\log b} = \frac{\log b}{\log c} \end{array}$
$∴ \log_{b} a = \log_{c} b [∵ \frac{\log_{e} x}{\log_{e} y} = \log_{y} x]$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App