Question

Q9IF log x – log (x - 1) = 5 then x
2. 2. [ Refer from attachment please]
is equal to
a) 31/32
b) 32/31
c) 25
d) 24

(Please solve this question in detail!!!!) ​

Answer 1

$\rm{\log _2\left(x\right)-\log _2\left(x-1\right)=5}$

• Add $\rm{\log _2\left(x-1\right)}$ to Both Sides

$\to\rm{\log _2\left(x\right)-\log _2\left(x-1\right)+\log _2\left(x-1\right)=5+\log _2\left(x-1\right)}$

$\to\rm{\log _2\left(x\right)=5+\log _2\left(x-1\right)}$

• Apply log Rule : $\rm{a=\log _{b}\left(b^{a}\right)}$

$\to\rm{\log _2\left(x\right)=\log _2\left(32\right)+\log _2\left(x-1\right)}$

• Apply log Rule : $\rm{\log _{c}(a)+\log _{c}(b)=\log _{c}(a b)}$

$\to\rm{\log _2\left(x\right)=\log _2\left(32\left(x-1\right)\right)}$

• Apply log Rule : $\rm{\textrm { If } \log _{b}(f(x))=\log _{b}(g(x)) \textrm { then } f(x)=g(x)}$

$\to\rm{x=32\left(x-1\right)}$

$\to\rm{x=32x-32}$

• Subtract 32x from both Sides

$\to\rm{x-32x=32x-32-32x}$

$\to\rm{-31x=-32}$

• Divide both Sides by -31

$\to\rm{\displaystyle x=\frac{32}{31}}$