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. ln√x+3=1

Accepted Answer

Hey everyone in this video we're asked to solve a logarithmic equation. We're told to reject any X value that will make the original function undefined and to use a calculator to approximate our solution to three decimal places. The equation we're given is lawn of the square root of X plus six is equal to five. Okay? So we have one square root of X. Not the square root of six. Sorry the square root of X plus six equals five. Okay. Alright. So we can go ahead and move the six over. We're going to get lawn of the square root of X is equal to -1. Alright. So what we want to do first is we just want to simplify this lawn the square root of X. Ok. So we know that if we have the end route of X that we can just write that as X to the one over end. Okay, So long as the square root of X can be written Lawn X. To the one half We still have -1 on the right hand side. Okay? So we're using this exponent rule here. All right. Now when we're looking at our log rules, let's recall if we have an exponent inside of our log log base B of X. To the exponent K. Well, we can bring the exponent out and multiply it. So we're going to have K log base B of X. So in this case let's go ahead and use that the right Lawn X. To the one half as one half. Lawn of X equals minus one. Okay, Multiplying by two. We get lawn of X is equal to -2. All right. And now what we wanna do is we want to change this. So instead of having a log we just have numbers in terms to work with. We want to get rid of that log. So we can recall our log rule where we have log base B of X equals to a number A. Okay, we can rewrite that as B. To the A equals X. Okay. So the base of the log rhythm becomes the base of the exponent. In our case we have launched. So the base of that log rhythm is E. Hey we have the exponent -2. Okay. That's gonna be our a value here minus two. And then we have equals X. The value inside of our log rhythm. So there we have it. We have X equals E. To the minus two. We just want to rewrite this so that it looks like the same format as the answers. So we can rewrite if we have a negative exponents, we can put it in the denominator. So this is going to be one over e squared. Okay. And using our calculator, that's gonna give us 0.135. So our solution is here. And let's just double check that that is indeed a solution that the original function is defined in case we have lawn the square root of X. Ok, so we have lawn the square root of X. What we want when we're taking the natural log, we need the inside of it to be bigger than zero. Okay, So we need the square root of X to be bigger than zero. In order for that function to be defined. Well, where's the square root of X? Bigger than zero? Well, that's going to be when X is bigger than zero. Okay. Alright. So this equation is bigger than zero. So we are okay. And the answer will be answer B one over B squared. Thanks for watching everyone. I hope this video helped.

Key Concepts

Properties of Logarithms

Domain of Logarithmic Functions

Solving Logarithmic Equations

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