Question

Prove using good format: log(nb)x=(log(b)x)/(1+log(b)n)

Answer 1

Answer:

${{ log_{b} \left(x\right) }}\div( 1 + {{log_{b} \left(n\right)}})$  =  ${{ log_{b} \left(x\right) }}\div( {{log_{b} \left(b\right)}} + {{log_{b} \left(n\right)}})$

${{ log_{b} \left(x\right) }}\div( {{log_{b} \left(b\right)}} + {{log_{b} \left(n\right)}})$ =  ${{ log_{b} \left(x\right) }}\div( {{log_{b} \left(nb\right)}})$   =   ${{ log_{nb} \left(x\right) }}$

Step-by-step explanation:

i) $\log_{nb} x$   =  ${{ log_{b} \left(x\right) }}\div( 1 + {{log_{b} \left(n\right)}})$  =  ${{ log_{b} \left(x\right) }}\div( {{log_{b} \left(b\right)}} + {{log_{b} \left(n\right)}})$

ii) therefore simplifying i) we get

    ${{ log_{b} \left(x\right) }}\div( {{log_{b} \left(b\right)}} + {{log_{b} \left(n\right)}})$ =  ${{ log_{b} \left(x\right) }}\div( {{log_{b} \left(nb\right)}})$   =   ${{ log_{nb} \left(x\right) }}$

iii) Hence the equation is proved.