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

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. 6+2 ln x=5

Accepted Answer

Hey everyone in this problem we are asked to solve a logarithmic equation. Okay. And to approximate our solution to two decimal places if we need to. And the equation we're given is 13 plus eight Long X equals nine. Okay so rewriting what were given 13 plus eight times? Lawn of X equals nine. Okay. Alright. So when we have a longer term equation like this with numbers, the first thing we want to do is try to simplify the numbers. Okay see what that gives us and then work with the logs afterwards. Okay so we move the 13 over to the right hand side. We're gonna be left with eight. Lawn of X is equal to minus four, Dividing by eight. We get one of X is equal to -1 half. And now we have a log rhythm on the left hand side and a number on the right hand side. And we have a rule to deal with that. Okay so let's recall our log rhythm rule that tells us if we have log base B of X equals to a number A. This is equivalent to writing B to the exponent. A. Is equal to x. Okay so the base B of the log becomes the base of the exponent. Okay in this case we have lawn. So we have a longer than with base E. So we have e. To the -1 half is equal to X. So writing it. So we can read it more easily. X. is equal to E. to the -1 half. Now we have a negative exponents. So if we have some number A. To the negative end. Okay? We know we can write this as one over A. To the end. So using this we can get X equals one over E. To the one half. We also know that if we have a number to the exponent one over end, this is the same as writing the end route of A. Okay, so we can write this using this blue rule here we can write the X is equal to one over the square root of E. Okay. And we've done these simplifying steps. These last two steps to match with the answers were given. Okay, so we have one over E. And that's going to be 0.61. Okay, When we use our calculator and so our answer X is equal to one over the square root of E. Which is 0.61. That's answer. D. Thanks for watching. I hope this video helped see you in the next one.

Key Concepts

Logarithmic Functions

Domain of Logarithmic Functions

Solving Logarithmic Equations

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