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1 taking the number of compoundings per year to

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Unformatted text preview: 3−0.1x 3. We first note that we can rewrite the equation 9 · 3x = 72x as 32 · 3x = 72x to obtain 3x+2 = 72x . Since it is not convenient to express both sides as a power of 3 (or 7 for that matter) we use the natural log: ln 3x+2 = ln 72x . The power rule gives (x + 2) ln(3) = 2x ln(7). Even though this equation appears very complicated, keep in mind that ln(3) and ln(7) are just constants. The equation (x + 2) ln(3) = 2x ln(7) is actually a linear equation and as such we gather all of the terms with x on one side, and the constants on the other. We then divide both sides by the coefficient of x, which we obtain by factoring. (x + 2) ln(3) x ln(3) + 2 ln(3) 2 ln(3) 2 ln(3) x = = = = = 2x ln(7) 2x ln(7) 2x ln(7) − x ln(3) x(2 ln(7) − ln(3)) Factor. 2 ln(3) 2 ln(7)−ln(3) Graphing f (x) = 9 · 3x and g (x) = 72x on the calculator, we see that these two graphs intersect 2 ln(3) at x = 2 ln(7)−ln(3) ≈ 0.7866. 100 4. Our objective in solving 75 = 1+3e−2t is to first isolate the expo...
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