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Exponential Equations Solved by Logarithms (6.8)

Exponential Equations Solved by Logarithms (6.8). Finding the value of the exponent when b x = a. A little POD. Before, we solved equations like this: 10 2 = x. We found the argument . Or like this: x 2 = 49. We found the base . Today we find the exponent . Estimate the value of x:

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Exponential Equations Solved by Logarithms (6.8)

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  1. Exponential Equations Solved by Logarithms (6.8) Finding the value of the exponent when bx = a.

  2. A little POD Before, we solved equations like this: 102 = x. We found the argument. Or like this: x2 = 49. We found the base. Today we find the exponent. Estimate the value of x: 10x = 3 x needs to be between which two numbers? How could you guess an answer? Use your calculator to guess and check. What is the base? The argument?

  3. Exponential equations-- guess and check • 8x = 64 • 3x = 27 • 2x = ½ Getting the hang of it? • 64x = 8 Try this by thinking a minute first.

  4. Exponential equations-- common bases 64x = 8 How can we solve this using a common base? What happens if we change the problem? 64x = 4 64x = 16 8x = ¼ Keep looking for that common base.

  5. Exponential questions-- common bases Find the common base for these. 144x = 1728 512x = 4 Notice how we’ve been solving for the exponent in each of these.

  6. Exponential equations-- common bases If 3x = 32, then what do we know about x? If 5x-3 = 55, then how can we solve for x? If 5x-3 = 25, what do we do first? We can work each of these problems by using guess and check, or thinking it out a little. We can also do it by setting each side to a common base. Let me show you one of them.

  7. An easier way to solve for exponents What happens when we can’t find a common base easily? Use logs! If 10x = 3, then x = log10 3 In general, if x = 10y, then y = log10x = log x. So, if 5x = 12, then x = log5 12 In more general, if x = by, then y = logbx. See the pattern? Your calculator does logs base 10 very well.

  8. Solve our POD and others using logs now If 10x = 3, then x = log103 • 10x = 3 • 10x = 457 • 10x = 392

  9. Solve our POD and others using logs now Sometimes you have to use a little algebra with logs. • 5(10x) = 216 (Divide by the number in front of the base first!) • 10.6x = 200 • 10-x = 39 (Try this two ways.) • 15(10.3x) = 157 Check your answers!

  10. Handy tools Keep these log formulas handy for use: log 10 = 1 log 100 = 2 log 1000 = 3 log 10x = x so that log 103 = 3 (Test it on your calculators.)

  11. Try a little Change of Base How might you solve this equation when the base is 6, not 10? 6x = 152 We’re still solving for an exponent, so let’s look at logs. x = log6152 But your calculator doesn’t do base 6 logs. We have to set this up differently.

  12. Try a little Change of Base How might you solve this equation when the base is 6, not 10? 6x = 152 x = log6152 Now, we can divide and solve. x = (log 152)/(log 6) This is a short-cut, called the change of base formula. See how it works in the problem?

  13. Try a little Change of Base Try the same pattern with these problems: 5x = 13 7x = 49 (Check this one to see if it makes sense.)

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