2.8.1 MATHPOWER TM 12, WESTERN EDITION 2.8 Chapter 2 Exponents and Logarithms

In 1987, the world’s population was 5 billion and was increasing at a rate of 1.6% per annum. An equation for this, as a function of time is: P = 5(1.016) t In the study of exponents and logs, different bases are often used. It would simplify matters to use the same base every time. Mathematicians have found that using base e is very advantageous. Express this using base e: Let = e k.By definition, k = ln ln = = (to 2 significant digits) Hence, = e The equation can now be written as: P = 5e 0.016t Growth Rate Initial population The exponent of base e is the growth rate

If 1987, the world’s population was 5 billion and was increasing at a rate of 1.6% per annum. Predict the world population in P = 5e 0.016t t = 27 years P = 5e x 27 P = 7.70 Therefore, the projected world population would be 7.7 billion people Population Growth

Light passes through glass with an absorption coefficient of Find the thickness of glass that will reduce the light’s intensity to half its initial intensity. I(t) = I o e -0.25t t = thickness of the glass I = intensity I o = initial intensity 50 = 100e -0.25t 0.5 = e -0.25t Take the natural logarithms of both sides: ln 0.5 = -0.25t t = 2.77 Therefore, the glass would be 2.8 mm thick An Application of Natural Logs

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