Solve each logarithmic equation in Exercises 49–92. Be sure to...

Question

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log₅ x+log₅(4x−1)=1

Accepted Answer

Hey everyone welcome back in this problem. We are asked to solve a logarithmic equation. Told to reject any X value that will make the original function undefined. And to approximate our solution to three decimal places. The equation we're given is log base two of x plus log base two of six. X minus five equals two. Alright so rewriting what we're given log base two of X plus log base two of six, X minus five equals two. All right. What we want to do with a problem like this is we're gonna combine those logarithms into one. That's going to make it easier to solve this equation. Ok. So let's recall the log rhythm role when we have two logarithms with the same base being added. If we have logarithms, Base B. Of A. Plus long rhythm base B of K. That's just gonna be longer than base B. Of A times K. So using this value or this rule we can rewrite what we're given as log the basis to for each of these. So log base two and then we have X times X. Whoops X times six. X -5. And on the right hand side we still have to. Alright, so we've managed to rewrite our equation to have a single log rhythm. Okay so it's easier to manipulate and now we have log rhythm equals a number. Let's recall the rule for dealing with that. If we have log rhythm base B of X equal to some number A. We can rewrite this as B. To the exponent A. Is equal to X. Okay those are equivalent expressions. So the base of the logarithms becomes the base of the exponent. Okay, so using this rule, What we can find is we get to to the exponent two. Okay, so the exponents going to come from the right hand side in the base. It's too And everything inside the log rhythm is going to go on the right hand side. So we're gonna have X times six X minus five. Alright, so expanding, we have four is equal to six, X squared minus five X. And moving the 4/0 is equal to six, X squared minus five, X minus four. Alright, if we're looking at this we can see that we can factor this, we get the factors three X -4 many times two X plus one. Now, if you didn't see that right away or if you're you know stressed in a test and you're you're not finding those factors right away, you can use the quadratic formula. Okay, use the quadratic formula with the values of this equation and you'll get the same the same routes. Okay, Alright, so using this to find the roots, we're going to get the roots of 4/3. Okay, from this first term we're going to the root minus one half from the second term. Alright, so these are the potential solutions we have we need to make sure that the original function is to find where these solutions are. So in the original function we have two things we need to worry about. We need the log base two and let me write on the right hand side so we don't run out of room. We need log base two of X to be okay. Okay, when we look at this and we look at the domain, we know that we need X bigger than zero. Okay? We can't take the logarithm of zero or something negative. So we need X bigger than zero. Well, 4/3 is bigger than zero. So that solutions. Okay, But minus one half is not. Okay, so the function, the original function will be undefined at minus one half. So that is not a solution. The second thing we need to check is that log base two of 6, X -5 is also going to work out. Okay, so the domain here, similarly, whatever is inside the log rhythm needs to be bigger than zero. So in this case we get six X minus five needs to be bigger than 06. X needs to be bigger than five. We need x bigger than 56. So we have X equals four thirds. That's bigger than one. We need to make sure it's bigger than 56, which is less than one. So it is indeed bigger than 56. So that solution is Okay. Alright, so we have X equals four thirds is our solution. Okay, if we approximate that with our calculator, we're gonna get 1.333. So we have answer a. I hope this video helped everyone. Thanks for watching.

Key Concepts

Properties of Logarithms

Domain of Logarithmic Functions

Solving Exponential Equations

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