Scientific calculators typically use Taylor series to produce the decimal expansions of logarithms. In this talk, I show how to compute logarithms to surprisingly high precision --- as many as ten decimal places --- by instead creatively using the Laws of Logarithms. This technique evolved from a Precalculus classroom demonstration, which will also be presented, that I've used to deepen both students' proficiency with the Laws of Logarithms as well as their basic numeracy with logarithms, such as estimating $\log_{10} 65,085$ without a calculator. Several of my former students, who are now secondary teachers, have successfully replicated this demonstration with their current students. Time permitting, I'll talk about how I responded (successfully) to a student's challenge of estimating $\sqrt[19]{25,757}$ without a calculator.