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: 10 x = 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?

Exponential equations-- guess and check 1. 8 x = x = x = ½ Getting the hang of it? x = 8 Try this by thinking a minute first.

Exponential equations-- common bases 64 x = 8 How can we solve this using a common base? What happens if we change the problem? 64 x = 464 x = 16 8 x = ¼ Keep looking for that common base.

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

Exponential equations-- common bases If 3 x = 3 2, then what do we know about x? If 5 x-3 = 5 5, then how can we solve for x? If 5 x-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.

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

Solve our POD and others using logs now If 10 x = 3, then x = log x = x = x = 392

Solve our POD and others using logs now Sometimes you have to use a little algebra with logs. 1. 5(10 x ) = 216 (Divide by the number in front of the base first!) x = x = 39 (Try this two ways.) 4. 15(10.3x ) = 157 Check your answers!

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

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

Try a little Change of Base How might you solve this equation when the base is 6, not 10? 6 x = 152 x = log 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?

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