Use Quotient Rule: \(\frac{{x}^{a}}{{x}^{b}}={x}^{a-b}\).
\[1\times {({x}^{2-b})}^{c}{(\frac{{x}^{b}}{{x}^{c}})}^{a}{(\frac{{x}^{c}}{{x}^{a}})}^{b}=1\]
Use Quotient Rule: \(\frac{{x}^{a}}{{x}^{b}}={x}^{a-b}\).
\[1\times {({x}^{2-b})}^{c}{({x}^{b-c})}^{a}{(\frac{{x}^{c}}{{x}^{a}})}^{b}=1\]
Use Quotient Rule: \(\frac{{x}^{a}}{{x}^{b}}={x}^{a-b}\).
\[1\times {({x}^{2-b})}^{c}{({x}^{b-c})}^{a}{({x}^{c-a})}^{b}=1\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[1\times {x}^{(2-b)c}{({x}^{b-c})}^{a}{({x}^{c-a})}^{b}=1\]
Regroup terms.
\[1\times {x}^{c(2-b)}{({x}^{b-c})}^{a}{({x}^{c-a})}^{b}=1\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[1\times {x}^{c(2-b)}{x}^{(b-c)a}{({x}^{c-a})}^{b}=1\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[1\times {x}^{c(2-b)}{x}^{(b-c)a}{x}^{(c-a)b}=1\]
Regroup terms.
\[{x}^{c(2-b)}{x}^{(b-c)a}{x}^{(c-a)b}=1\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[{x}^{c(2-b)+(b-c)a+(c-a)b}=1\]
Use Definition of Common Logarithm: \({b}^{a}=x\) if and only if \(log_b(x)=a\).
\[c(2-b)+(b-c)a+(c-a)b=\log_{x}{1}\]
Expand.
\[2c-cb+ab-ac+bc-ba=\log_{x}{1}\]
Simplify \(2c-cb+ab-ac+bc-ba\) to \(2c-ac\).
\[2c-ac=\log_{x}{1}\]
Factor out the common term \(c\).
\[c(2-a)=\log_{x}{1}\]
Divide both sides by \(c\).
\[2-a=\frac{\log_{x}{1}}{c}\]
Subtract \(2\) from both sides.
\[-a=\frac{\log_{x}{1}}{c}-2\]
Multiply both sides by \(-1\).
\[a=-\frac{\log_{x}{1}}{c}+2\]
Regroup terms.
\[a=2-\frac{\log_{x}{1}}{c}\]
a=2-log(x,1)/c
