Positive exponential substitutions For a>0, a 1, the substitution u=ax always satisfies u>0. Also a2x=u2 and a-x= 1u. A positive-base exponential never equals zero or a negative number. Multiplication by u is safe because u 0; the quadratic may nevertheless have roots outside its allowed range. Choose one base; rewrite powers consistently; state u>0; solve the quadratic; retain positive roots; use x=log_a u, keeping exact values before rounding. Example For 9x-4(3x)+3=0, set u=3x>0. (u-1)(u-3)=0, giving x=0 or x=1. Check: a negative root for u is rejected, but a negative value of x may be valid. For example, 3x= 13 has x=-1.

For \(a>0\), \(a\ne1\), the substitution \(u=a^x\) always satisfies \(u>0\). Also \(a^{2x}=u^2\) and \(a^{-x}=\dfrac1u\).

A positive-base exponential never equals zero or a negative number. Multiplication by \(u\) is safe because \(u\ne0\); the quadratic may nevertheless have roots outside its allowed range.

Choose one base; rewrite powers consistently; state \(u>0\); solve the quadratic; retain positive roots; use \(x=\log_a u\), keeping exact values before rounding.

For \(9^x-4(3^x)+3=0\), set \(u=3^x>0\). \((u-1)(u-3)=0\), giving \(x=0\) or \(x=1\).

Check: a negative root for \(u\) is rejected, but a negative value of \(x\) may be valid. For example, \(3^x=\dfrac13\) has \(x=-1\).