Tables of logarithms and trigonometric functions were common in math and science textbooks. Mathematical and navigation handbooks included tables of the logarithms of trigonometric functions as well. Exponentiation has two inverse operations; roots and logarithms. Logarithms and exponentials with the same base cancel each other. See logarithmic units for logarithms taken in different bases. Similarly, factorials can be approximated by summing the logarithms of the terms. It allowed the display of expressions of fractions, exponents, logarithms, powers and square roots etc. Euler introduced the use of the exponential function and logarithms in analytic proofs. Some advanced players allow exponentiation, roots, logarithms, and other operations. Base-e (natural) logarithms and exponentiation can be used, but not base-10. The logarithm of a product is simply the sum of the logarithms of the factors. Take the natural logarithms of the equation. Euler also suggested that the complex logarithms can have infinitely many values. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms. Charles Babbage designed machines to tabulate logarithms and other functions in 1837.