Question

a) If x = log3/5, y = log5/4and z = 2log√3/2 , prove that 5^x+y-z= 1
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Answer 1

$\textbf{Given:}$

$x=\log\dfrac{3}{5}$

$y=\log\dfrac{5}{4}$

$z=2\log\dfrac{\sqrt{3}}{2}$

$\textbf{To find:}$

$5^{x+y-z}=1$

$\textbf{Solution:}$

$\text{Consider,}$

$x+y-z$

$=\log\dfrac{3}{5}+\log\dfrac{5}{4}-2\log\dfrac{\sqrt{3}}{2}$

$\text{Using power rule of logarithm}$

$=\log\dfrac{3}{5}+\log\dfrac{5}{4}-\log(\dfrac{\sqrt{3}}{2})^2$

$=\log\dfrac{3}{5}+(\log\dfrac{5}{4}-\log\dfrac{3}{4})$

$\text{Using quotient rule of logarithm}$

$=\log\dfrac{3}{5}+\log\dfrac{\frac{5}{4}}{\frac{3}{4}}$

$=\log\dfrac{3}{5}+\log\dfrac{5}{3}$

$\text{Using product rule of logarithm}$

$=\log(\dfrac{3}{5}{\times}\dfrac{5}{3})$

$=\log\,1$

$=0$

$\implies\,x+y-z=0$

$\implies\,5^{x+y-z}=5^0=1$

$\textbf{Answer:}$

$\boxed{\bf\,5^{x+y-z}=1}$.

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Answer 2

Step-by-step explanation:

here is your step by step explanation