Question

Suppose that $4^{x_1}=5,5^{x_2}=6,6^{x_3}={\mathrm{7.....127}}^{x_{124}}$ Find $x_1{x}_2{x}_3......x_{124}$

• A. $2$
• B. $\frac{5}{2}$
• C. $3$
• D. $\frac{7}{2}$

Answer 1

The correct option is D $\frac{7}{2}$

$4^{x_1}=5$

Apply logarithm on both sides

$log{4}^{x_1}=\log 5$

$x_{1}log4=log5$

$x_1=\frac{log5}{log4}$

$[sincelog{a}^n=nloga]$

Similarly $x_2=\frac{log6}{log5}$

$x_3=\frac{log7}{log6}$

................

$x_{123}=\frac{log127}{log126}$

$x_{124}=\frac{log128}{log124}$

So $x_1\times x_2\times x_3\times x_4\times x_5.....x_{123},x_{127}$

$=\frac{log5}{log4}\times \frac{log6}{log5}\times \frac{log7}{log6}\times ...........\times \frac{log127}{log126}\times \frac{log128}{log127}$

Notice that each numerator except the last one cancells out with the denominator of the next term.

$=\frac{log128}{log4}$

$=\frac{log{2}^7}{log{2}^2}$

$=\frac{7log2}{2log2}$

$=\frac{7}{2}$