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. 2 log₃(x+4)=log₃ 9 + 2

Accepted Answer

everyone in this video. We're asked to solve the following logarithmic equation. We're told to reject any X values that make the original function undefined. And to round the or approximate the solution to three decimal places. The equation we're given is five log base five of X plus one is equal to log base five of 125 plus two. Okay so writing out what we have five log base five of X plus one is equal to log base five of 125 plus two. Okay now before dealing with the left hand side on the right hand side we have just numbers. Okay so let's try to work with the numbers first um to see if anything kind of comes from that. That's typically the easier route to work with. So we work with those first and then see what we can do with the rest. So on the left we're gonna leave it alone. Five log base five of X plus one is equal to log base five of 125. While we're looking for a number where we go five to the exponents what equals 100 and 25. Well in this case that's three. Okay so we're just gonna have three Plus two on the right hand side. Okay simplifying five log base five of X plus one is equal to five. Okay now we can divide by five so we don't need to worry about using our log rules to deal with this five. We can just simply divide. Okay so log base five of X plus one is equal to one. Alright, so now let's recall the log rule, we have log base B of some number X. Or some value X equals to A. And we know that we can write that as B to the A equals x. Okay, so the base of the log becomes the base of the exponent. So in our case the base of the log is five. We take to the exponent one and everything in the log that's what it's going to be equals two. Okay, so five to the exponent one is equal to X plus one. Okay. Moving the X or sorry? Moving the one to the left hand side. We have X is equal to four. Alright, so there's a potential solution. Now let's just double check that the original function is defined there. Okay, so we only have one term we need to double check with here. We have this log base five of x plus one. Okay. And its domain? Well we can't take the log rhythm of anything zero or less. So we need X plus one to be bigger than zero which tells us that X needs to be bigger than minus one. Okay, in this case X equals four is bigger than minus one. So we get a check mark, we pass that and that is indeed a solution. Okay, so X equals four is a solution. Okay And now the solutions are also asking us where the function is undefined. Okay, well we already looked at the domain, so the function is defined for X bigger than minus one, which means that it's going to be undefined If X is less than or equal to -1. Okay. Alright. So that's going to be answer D. The function is undefined less than or equal to minus one, and we have the solution of X equals four. Thanks everyone for watching. See you in the next video.

Key Concepts

Properties of Logarithms

Domain of Logarithmic Functions

Converting Logarithmic Equations to Exponential Form

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