A solution that emerges from the process of solving a problem but is not a valid solution to the original problem.

A function that reverses the effect of another function. If f(a) = b, then f⁻¹(b) = a.

The inverse function of an exponential function, expressing the power to which a base must be raised to produce a given number.

A function in which the independent variable appears in the exponent.

Exponential: Isolate exponential term, take logarithm. Logarithmic: Isolate log term, convert to exponential form. Extraneous solutions are more common in logarithmic equations.

Exponential: Domain is all real numbers. Logarithmic: Domain is positive real numbers (argument must be > 0).

$\log_b(MN) = \log_b(M) + \log_b(N)$

$\log_b(M/N) = \log_b(M) - \log_b(N)$

$\log_b(M^p) = p \cdot \log_b(M)$

$\log_b(M) = \frac{\log_c(M)}{\log_c(b)}$

$f(x) = ab^{x-h} + k$

$f(x) = a \log_b(x-h) + k$

$f^{-1}(x) = h + \frac{\ln((x-k)/a)}{\ln(b)}$

$f^{-1}(x) = b^{\frac{(x-k)}{a}} + h$