36 MATHEMATICAL THEORY OF INVESTMENT 15. Exponential equations. In most of the equations that occur in elementary algebra the exponents are known and the number affected by the exponent is unknown; for example, xt+ 5 a- +6=0. It was noted in § 2 that when the number of terms of a geometrical progression is unknown, it is ordinarily impossible to find it by elementary means, since it occurs as an exponent. For example, if the first term, the last term, and the ratio of a geometrical progression are 5, 98415, and 3, respectively, equation ((7) of § 2 gives 98415= 5 (3)"-1, where n, the unknown, is a part of the exponent. This equation is of the same type as the equation <f:=y and may be treated in the same way. An equation in which the unknown number occurs as an ex- ponent is called an exponential equation. The simplest form of an exponential equation is given by the - equation 4 a^b. (1) This equation can be solved readily by taking logarithms of both sides. By the theorem for the logarithm of a power, x log a = log b; whence x = °^ f2') log a v ' It must be carefully noted that the expression on the right side of equation (2) is the quotient of two logarithms and not the log- arithm of a quotient, so that it cannot be written as log b — log a. No general rule for the solution of exponential equations can be given, but the solutions of two of the following examples will serve to illustrate the method to be used in the simpler cases. LOGARITHMS37
EXAMPLES 1. Solve the equation 2°' = 3. Solution. Taking logarithms of both sides, we find x log 2 = log 3, and solving this linear equation for a", log3. -log2' or, using a four-place table, x = '4771 . .3010 By actual division (which may of course be performed by logarithms) ;!;=:1.685+. 2. Solve a?^2 x lo3—3 = Wx+\ for x. Solution. Taking logarithms of both sides, we have {x + 2) log 37 +(a; - 3) log 15 = (2 x + 1) log 14, or x log 37 + x log 15 - 2z log 14 =- 2 log 37 +3 log 15 + log 14. Solving fora;, we find ,, _ - 2 log37+31ogl5 + logl4 log 37 + log 15—2 log 14 _ - 3.1364 + 3.5283 + 1.1461 - 1.5682 + 1.1761 - 2.2922 _ 1.5380 - .4621 = 3.402+. 3. Find x from the equation SO" = 3000. \ 4. Given 3^^2t= 29; find x. 5. Given 2"- 3v = 2000 and 3 y = 5 x; find x and y. 6. Given ; = ar"-1; find n. 7. Given the two equations I = ar"-1, rl - a .»=——-; r — 1 find n in terms of a, I, and s. 8. Find the expression for n from the equation .<? = PCl +!')». 9. Find n from the equation A =(1 + ')" — 1 _