Putting these two steps together we simplify ln e 2 t ln 1 9 to 2 t ln9 We

# Putting these two steps together we simplify ln e 2 t

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Putting these two steps together, we simplify ln ( e - 2 t ) = ln ( 1 9 ) to - 2 t = - ln(9). We arrive at our solution, t = ln(9) 2 which simplifies to t = ln(3).(Can you explain why?)The calculator confirms the graphs off(x) = 75 and 1 . 099. y = f ( x ) = 9 · 3 x and y = f ( x ) = 75 and y = g ( x ) = 7 2 x y = g ( x ) = 100 1+3 e - 2 x 5. We start solving 25 x = 5 x + 6 by rewriting 25 = 5 2 so that we have ( 5 2 ) x = 5 x + 6, or 5 2 x = 5 x + 6. Even though we have a common base, having two terms on the right hand side of the equation foils our plan of equating exponents or taking logs. If we stare at this long enough, we notice that we have three terms with the exponent on one term exactly twice that of another. To our surprise and delight, we have a ‘quadratic in disguise’. Letting u = 5 x , we have u 2 = (5 x ) 2 = 5 2 x so the equation 5 2 x = 5 x + 6 becomes u 2 = u + 6. Solving this as u 2 - u - 6 = 0 gives u = - 2 or u = 3. Since u = 5 x , we have 5 x = - 2 or 5 x = 3. Since 5 x = - 2 has no real solution, (Why not?) we focus on 5 x = 3. Since it isn’t convenient to express 3 as a power of 5, we take natural logs and get ln (5 x ) = ln(3) so that x ln(5) = ln(3)
7.3 Exponential Equations and Inequalities 365 or x = ln(3) ln(5) . When we graph f ( x ) = 25 x and g ( x ) = 5 x + 6, we see that they intersect at x = ln(3) ln(5) 0 . 6826. 6. At first, it’s unclear how to proceed with e x - e - x 2 = 5, besides clearing the denominator to obtain e x - e - x = 10. Of course, if we rewrite e - x = 1 e x , we see we have another denominator lurking in the problem: e x - 1 e x = 10. Clearing this denominator gives us e 2 x - 1 = 10 e x , and once again, we have an equation with three terms where the exponent on one term is exactly twice that of another - a ‘quadratic in disguise.’ If we let u = e x , then u 2 = e 2 x so the equation e 2 x - 1 = 10 e x can be viewed as u 2 - 1 = 10 u . Solving u 2 - 10 u - 1 = 0, we obtain by the quadratic formula u = 5 ± 26. From this, we have e x = 5 ± 26. Since 5 - 26 < 0, we get no real solution to e x = 5 - 26, but for e x = 5 + 26, we take natural logs to obtain x = ln ( 5 + 26 ) . If we graph f ( x ) = e x - e - x 2 and g ( x ) = 5, we see that the graphs intersect at x = ln ( 5 + 26 ) 2 . 312 y = f ( x ) = 25 x and y = f ( x ) = e x - e - x 2 and y = g ( x ) = 5 x + 6 y = g ( x ) = 5 The authors would be remiss not to mention that Example 7.3.1 still holds great educational value. Much can be learned about logarithms and exponentials by verifying the solutions obtained in Example 7.3.1 analytically. For example, to verify our solution to 2000 = 1000 · 3 - 0 . 1 t , we substitute t = - 10 ln(2) ln(3) and obtain 2000 ? = 1000 · 3 - 0 . 1 - 10 ln(2) ln(3) 2000 ? = 1000 · 3 ln(2) ln(3) 2000 ? = 1000 · 3 log 3 (2) Change of Base 2000 ? = 1000 · 2 Inverse Property 2000 X = 2000 The other solutions can be verified by using a combination of log and inverse properties. Some fall out quite quickly, while others are more involved. We leave them to the reader.
366 Exponential and Logarithmic Functions Since exponential functions are continuous on their domains, the Intermediate Value Theorem 4.1 applies. As with the algebraic functions in Section ?? , this allows us to solve inequalities using sign diagrams as demonstrated below.
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