In Exercises 64–73, solve each exponential equation. Where necess...

Question

In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 8^x = 12143

Accepted Answer

Hello. Today we're going to be solving an exponential equation for X. And we're going to represent its solution in terms of common logarithms. So the equation given to us is five raised to the power of x equal to 91,325. Now, in order to bring down this X from the exponent, we can go ahead and multiply both sides of this equation by the common logarithms. Now keep in mind the common logarithms has a base of 10 but notation wise it's not necessary to write it here. So multiplying both sides of the equation by the common logarithms is going to give us the logarithms of five to the power of X equal to the logarithms of 91,325. Now there's a logarithmic law that allows us to bring down an exponent from inside the argument. So suppose we have log of the argument X to the power of N. We could bring n in front of the logarithms as a product and represent this as N times log of X. And that's what we're gonna do on this left hand side. I'm gonna go ahead and take X and bring it forward in front of the logarithms. Using our exponential rule for logarithms. Doing so is going to give me X times the logarithms of five equal to the logarithms of 91,325. Now I want to go ahead and sell for X. And in order to do that, I'm gonna divide both sides of this equation by the logarithms of five. Doing so is going to give me X equal to the longer rhythm of 91, over the logarithms of five. And this is gonna be the solution for X. Now the problem also wants us to represent the solution as a decimal rounded to the nearest 100. So if we plug this value into a calculator and approximate it to the nearest 100 we get an approximate value of 7.10. And this is gonna be our total solution to this problem. And that also means that the answer to this problem is going to be D. So I hope this video helps you in understanding how to use the common law algorithm and I'll go ahead and see you all in the next video.

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