481 4.4 Evaluating Logarithms and the Change-of-Base Theorem SOLUTION (a) If A = 0, then A K = 0 and the equation is as follows. t = 11.26 * 1092 ln A1 + 8.33AA KB B ln 2 Given formula t = 11.26 * 1092 ln 1 ln 2 A K = 0, so ln 11 + 02 = ln 1 t = 11.26 * 1092102 ln 1 = 0 t = 0 The rock is new (0 yr old). (b) Because A K = 0.212, we have the following. t = 11.26 * 1092 ln 11 + 8.3310.21222 ln 2 Substitute. t ≈1.85 * 109 Use a calculator. The granite is about 1.85 billion yr old. S Now Try Exercise 77. LOOKING AHEAD TO CALCULUS The natural logarithmic function ƒ1x2 = ln x and the reciprocal function g1x2 = 1 x have an important relationship in calculus. The derivative of the natural logarithmic function is the reciprocal function. Using Leibniz notation (named after one of the co-inventors of calculus), we write this fact as d dx 1ln x2 = 1 x . EXAMPLE 7 Modeling GlobalTemperature Increase Carbon dioxide in the atmosphere traps heat from the sun. The additional solar radiation trapped by carbon dioxide is radiative forcing. It is measured in watts per square meter 1W/m22. In 1896 the Swedish scientist Svante Arrhenius modeled radiative forcing R caused by additional atmospheric carbon dioxide, using the logarithmic equation R = k ln C C0 , where C0 is the preindustrial amount of carbon dioxide, C is the current carbon dioxide level, and k is a constant. Arrhenius determined that 10 … k … 16 when C = 2C0 . (Data from Clime, W., The Economics of Global Warming, Institute for International Economics, Washington, D.C.) (a) Let C = 2C0. Is the relationship between R and k linear or logarithmic? (b) The average global temperature increase T (in °F) is given by T1R2 = 1.03R. Write T as a function of k. SOLUTION (a) If C = 2C0 , then C C0 = 2, so R = k ln 2 is a linear relationship, because ln 2 is a constant. (b) T1R2 = 1.03R T1k2 = 1.03k ln C C0 Use the given expression for R. S Now Try Exercise 75. Logarithms with Other Bases We can use a calculator to find the values of either natural logarithms (base e) or common logarithms (base 10). However, sometimes we must use logarithms with other bases. The change-of-base theorem can be used to convert logarithms from one base to another.
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