One-to-One Property of Exponents: For any algebraic expressions \(S\) and \(T\), and any positive real number \(b \neq 1\), \(b^S = b^T\) if and only if \(S=T\).
Common Mistakes
Incorrectly Applying the One-to-One Property of Exponents: Attempting to equate exponents when the bases are not the same, without first rewriting them to have a common base.
Errors in Using Logarithms to Solve Exponential Equations: Incorrectly applying logarithm laws, especially the power rule (e.g., \(\ln(5^{x+2}) = (x+2)\ln(5)\), not \(x\ln(5)+2\)).
Forgetting to Check for Extraneous Solutions in Logarithmic Equations: Failing to verify that all solutions obtained make the arguments of the original logarithms positive.
Errors in Using the One-to-One Property of Logarithms: Not first condensing logarithmic expressions on each side of the equation to a single logarithm before equating arguments.
Treating \(\ln(A-B)\) as \(\ln(A)-\ln(B)\) or \(\frac{\ln(A)}{\ln(B)}\): Incorrectly splitting logarithms of differences or quotients of logarithms. (Common algebraic errors with log laws)
Solving Equations Quadratic-in-Form Incorrectly: Forgetting to substitute back after solving for \(u\) in an equation like \(e^{2x}-e^x-56=0\), or incorrectly solving the resulting exponential/logarithmic equations after substitution.