Math Algebra • Exponential and Logarithmic Functions
Use the format log_3(x) for base-3 logs, ln(x) for natural logs, and log(x) for logs that use the selected default base. Use parentheses for arguments and ^ for exponents.
Condensing combines sums and differences of logs into one log using product and quotient rules. Expanding does the reverse by splitting a single log of a product, quotient, or power into separate terms.
It uses log_b(UV) = log_b(U) + log_b(V), log_b(U/V) = log_b(U) - log_b(V), and log_b(U^k) = k log_b(U) when the expressions are defined. It also simplifies common values like log_b(1) = 0 and log_b(b) = 1.
A logarithm requires a base b > 0 and b != 1, and its argument must be greater than 0. If an argument is not positive for some x values, the expression is undefined there and the graph and sample check reflect that.
It rewrites log_b(A) in an equivalent form such as ln(A)/ln(b) so you can compare or standardize expressions across different bases. This is useful when mixing ln(x) with log(x) or custom bases.