What Does x Equal?
Date: 01/16/98 at 18:15:44 From: Amanda Hoffman Subject: Algebra 2/Trigonmetry Dr. Math, We have been doing problems with logarithms. Can you help me with this problem? 32 times 4 to the 2x power equals 12. I have to know what x equals. Also, what is a logarithm? Amanda
Date: 01/21/98 at 22:59:19 From: Doctor Wolf Subject: Re: Algebra 2/Trigonmetry Hi Amanda, To answer your last question first ... a logarithm IS an exponent. There is an excellent article in the ARCHIVES entitled "Hints About Logs." It would be good for you to review this short document on logs and their properties often as you progress in your class: http://mathforum.org/dr.math/problems/loghints.html Being able to convert logarithmic equations to exponential equations and back is essential to solving log problems. Back to the problem you presented: 32*4^(2x) = 12 Dividing both sides of this equation by 32 we have 4^(2x) = 12/32 Let's do a little work on the left side of this equation first. 4^(2x) = (4^2)^x = 16^x using the multiplication property of exponents and the fact that 4^2(squared) equals 16. Now for the right side: 12/32 = 6/16 I reduced the fraction, but wanted 16 in the denominator for a reason which will become clear in the next few steps. Therefore: 16^x = 6/16 using the new left and right sides from above. But this means that x must be the log of 6/16 to the base 16, by the definition of a logarithm. We can go just a bit further: Since log (a/b) = (log a) - (log b) (division prop. of logs) log (6/16)(base 16) = log 6 (base 16) - log 16 (base 16) = log 6 (base 16) - 1 Why? Finally: x = log 6 (base 16) - 1 I hope this helped. It's not really what I would call a trivial problem, and may have various forms for an answer. Don't hesitate to get back to me if you have further questions. -Doctor Wolf, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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