Solve each equation. Give solutions in exact form. See Examples 5...

Question

Solve each equation. Give solutions in exact form. See Examples 5–9. log x^2 = (log x)^2

Accepted Answer

Hello, everybody. I have everything. All right today, then we're gonna look at this question that's asking us to solve the logarithmic equation or X and provide an exact solution where our equation is. And I'll write it down as we go two, multiplying the log base four of X to the power of two equal to the log base four of X. And that whole thing including the log raised to the power of two. Now our answers show up as a being one comma 256 B, 100 and 28 C one comma 64 D 64. So let's just work through this little by little step by step to get to our final answer. Now, the first thing we can see when we look at the left side of this equation, we can remember. So, or, or recall some different logarithm rules that we have. In this case, we see that the X inside of the logarithm is raised to the power of two. So we can look at our power rule which I'm just gonna write a general format for, let's say that log base B of A is raised to the power of C that can be rewritten as C multiplying the log base B of A. So applying that rule to the equation that we have, we would move the exponent two over in front of the equation to multiply. And I'm just gonna rewrite that as two multiplying two of log base four of X. And notice it's just X now not X to the power of two because like I mentioned, we move that power over to the front and that will continue to be equal to the log base four of X with the whole thing to the power of two. This side, we cannot move the exponent forward because it's not just the X being raised to that power, it's the entire side of the equation. So it has to stay like that. Now we know that two multiplying two is four. So it, that side of the equation will turn into four log base four of X equal to log base four of X with that whole thing raised to the power of two. Now, we can do some simplification. Now, the way that I'm going to do it is by use substitution. And I'm just gonna write this here, let you be equal to log base four of X. Another way that you could do this is you see this and you think OK, the right side is log based four of X entirely raised to the power of two. And on the left side, I have log base four of X as well. So you could technically divide both sides by log based four of X to simplify a little bit. However, if you decide to do this, you need to keep in mind that there is a restriction for this equation. For example, let's say if X is equal to one that's gonna be equal to zero and you can't divide by zero, therefore X is equal to one will be a restriction. So if you decide to take that route, you need to keep that restriction in mind because if you don't, you will only get one answer. And for this case, there will be two. Now if you do use substitution, like I will walk you through. Now you don't have to keep that restriction in mind because it will take it into account as you go. So I'm just gonna do that now. And like I said, let U equal to log base form. So that means we can rewrite the equation that we have as four U is equal to us to the power of two. So now we can move the four U over and we would have zero is equal to U to the power two minus four U. And now by doing some factoring, we can do zero is equal to U multiplying U minus four. And now we know by looking at this, you must be equal to one to zero. I'm sorry to zero and four positive four. Now, using that information, we can do plug some plugging in back to the equation that we had. And we can have uh since we said U is equal to log with four S one of our U, let's say we're gonna start with zero, we have zero is equal to log base four of X. Now, this is just a simple algorithm equation that we can turn into exponent equations. So let's go back to our recall. And let's say we have something like log base B of A is equal to C that can be rewritten as an exponential equation as B raised to the power C is equal to A. So using that information, we can go back to what we have zero is equal to log base Forex and rewrite it as an exponential equation as four raised to the power of zero is equal to X. Now we know race to the four race to the power of zero, any number raised to the power of zero is one. So we'll have our first answer as X is equal to one and that will be our first answer. And now we can go to our second answer. We have our second U which was four, we can plug that in as well and have four is equal to log base four of X. Just plugging that into the equation that we had U equals log base four. Of X. So our U is four in this case and we do the same thing using our recall, we can raise four to the power of four is equal to X and four. Race to the power of four is 256. So X will be equal to 56. And that will be our second answer for this question. And that'll give us our final answer of one comma or letter A in this case. So I hope that was helpful. And until next time.

Solve each equation. Give solutions in exact form. See Examples 5–9. log x^2 = (log x)^2

Key Concepts

Logarithmic Properties

Quadratic Equations

Exact Solutions

Watch next