what is value of x in an arithmetic progression involving logarthms $\log 2,\log 2^{x-1}$, and $\log 2^{x+3}$ are $3$ consecutive terms of an arithmetic progression; find  (i) the value of $x$;

If $\log 2,\log 2^{x-1}$, and $\log 2^{x+3}$ are three consecutive terms of an arithmetic progression, there must be a common difference $d$ such that $\log 2^{x-1}=\log 2+d$ and $\log 2^{x+3}=\log 2^{x-1}+d=$  $\log 2+2d$. From basic properties of the logarithm we know that $$\log 2^{x-1}=(x-1)\log 2$$ and $$\log 2^{x+3}=(x+3)\log 2\;,$$ so $$\left\{\begin{align*}&(x-1)\log 2=\log 2+d\\&(x+3)\log 2=\log 2+2d\;.\end{align*}\right.$$ This system can easily be solved for $d$ and $x$.
