Solve for x: Natural Log Help!

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Homework Help Overview

The problem involves solving the equation ln(x+10) + ln(x+4) = 2, which falls under the subject area of logarithmic equations. Participants are exploring the steps to isolate x and the implications of their algebraic manipulations.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the transformation of the logarithmic equation into a quadratic form and the application of the quadratic formula. There are questions about the validity of certain steps and the implications of the solutions derived from the quadratic formula.

Discussion Status

The discussion includes various attempts to simplify the equation and clarify the steps taken. Some participants express uncertainty about the next steps, while others provide insights into the simplification process. There is a recognition of the need to ensure that the solutions remain valid within the context of the logarithmic functions involved.

Contextual Notes

Participants note the importance of ensuring that the solutions do not lead to negative arguments for the logarithmic functions, as this would invalidate the solutions. There is also mention of specific values derived from the quadratic formula that may not satisfy the original equation.

tommy2st
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natural log helpp!

Homework Statement


ln(x+10)+ln(x+4)=2



2. The attempt at a solution

okay so i took it down to (-14+-(14^2 - 4(e^2 - 40))^1/2) / 2

which i broke down to 356 - 4e^2 inside the sqrt.

am i on the right track and if so Whats the next step?
 
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looks ok, I'm not sure what you mean by next step though
x = (-14+-(14^2 - 4(e^2 - 40))^1/2) / 2 is your answer
 


wukunlin said:
looks ok, I'm not sure what you mean by next step though
x = (-14+-(14^2 - 4(e^2 - 40))^1/2) / 2 is your answer

it can be broken down further into something more simplified.

the example i have goes down from x^2 + 4x -(e^2 + 5)=0

to x= -2+-sqrt 9+e^2

but it doesn't show the steps used
 


i figured it out. the answer was -7+ (9+e^2)^1/2
 


ln(x+10)+ln(x+4)=2
ln[(x+10)(x+4)] = 2
(x+10)(x+4) = e^2
x^2 + 14x + 40 = e^2
x^2 + 14x + 40-e^2 = 0
x = ( -14 +- sqrt((14^2)-4*1*(40-e^2)) ) / (2) [quadratic formula]
x = answer
 


legendary_ said:
ln(x+10)+ln(x+4)=2
ln[(x+10)(x+4)] = 2
(x+10)(x+4) = e^2
x^2 + 14x + 40 = e^2
x^2 + 14x + 40-e^2 = 0
x = ( -14 +- sqrt((14^2)-4*1*(40-e^2)) ) / (2) [quadratic formula]
x = answer

No. if you take the negative in the "+-", you get a number less than -10 so that both x+10 and x+ 4 are negative and you cannot take the logarithm.

Only ( -14 + sqrt((14^2)-4*1*(40-e^2)) ) / (2) satifies the original equation.
 

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