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. 10^x = 7000

Accepted Answer

Hello. Today we're going to be solving an exponential equation for X. And we're going to represent its solution using common logarithms. So the exponential equation given to us is to raise to the power of X equal to 3000. Now what I want to do is go ahead and bring this X forward in front of the two In order to represent it as a product instead of an exponent. But how do we do that? Well, in order to do that, I can multiply both sides of this equation by the common logarithms. Now keep in mind the common logarithms has a base of 10 but you know, it's not necessary to write it here. So multiplying both sides of this equation by the common logarithms is going to give me the logarithms of two to the power of x equal to the logarithms of 3000. Now there's an exponential rule for logarithms which states that if we have a logarithms with an argument raised to a power, we can bring that power in front of the logarithms as a product. Thus we are allowed to rewrite this logo rhythm as n times the logarithms of X. And that's what we're gonna do on this left hand side. We're going to bring this power of X in front of the logo rhythm as a product. Doing so is going to give me X times the logarithms of two equal to the logarithms of 3000. Now again we want to go ahead and solve for X. And in order to do that, we're gonna divide both sides of this equation by the logarithms of two. Doing so is gonna give me x equal to the logarithms of 3000, Divided by the logarithms of two. And this is going to be the solution to this exponential equation. Now keep in mind the problem also wants us to express this as a decimal rounded to the nearest 100th. So if we plug this into a calculator and approximate it to the nearest 100th place, we're gonna get an approximate value of 11.55 and this is gonna be the total solution to this equation. That also means that the answer to this problem is going to be C. So I hope this video helps you in understanding how to use common logarithms and I'll go ahead and see you all in the next video.

Watch next