Logarithmic Equations and Substitution

For a base \(a>0\), \(a\ne1\), when both \(a^{2x}\) and \(a^x\) occur, let \(u=a^x\), so \(u>0\). Solve the polynomial in \(u\), reject non-positive values and then recover \(x\).

For a logarithmic equation, record the positive-argument restrictions first. Combine logarithms with matching bases, solve the resulting equation and test every candidate in the original domain. Squaring or clearing factors can introduce inadmissible roots.

A graphical solution is an intersection of the two sides or a zero of their difference. Choose a window that respects the domain and check all visible branches; a graph supports a numerical answer but does not replace exact algebra when an exact value is requested.