Logarithim Problem

4 months ago

Gimme a sec

4 months ago

$\ln (x + 2) - \ln x = \ln (x + 5)$

4 months ago

4 months ago
Vocaloid:

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4 months ago
Vocaloid:

log rule 2, combine the ln(x+2) and the ln(x) on the left into one ln

4 months ago

$\ln \frac{ x + 2 }{ x } = \ln (x + 5)$

4 months ago
Vocaloid:

good, now when you raise both sides to the power of e you just get (x+2)/x = x + 5 then it's just algebra to solve for x

4 months ago
Vocaloid:

let me know what you get

4 months ago

Sorry. Got pulled away. $\frac{ x + 2 }{ x } = x + 5$ $x + 2 = x^2 + 5x$ $x^2 + 4x - 2 = 0$ Quadratic Formula $\frac{ -4 \pm \sqrt (4^2 -4(1)(-2) }{ 2(1) }$ $\frac{ -4 \pm \sqrt24 }{ 2 }$ $\frac{ -4 \pm 2 \sqrt 6 }{ 2 }$ $-2 \pm \sqrt 6$

4 months ago
Vocaloid:

good, that's what I end up getting too

4 months ago
Vocaloid:

I believe you also have to check for extraneous solutions though

4 months ago
Vocaloid:

so you would plug in -2 + root(6) and -2 - root(6) into both equations to make sure both ln terms are defined

4 months ago

Oh yeah

4 months ago

Only the positive one works for me.

4 months ago
Vocaloid:

yup that's what I got too, gj

4 months ago

If I may ask, how did you check for extraneous solutions.

4 months ago
Vocaloid:

we have an ln(x) term so x must be positive

4 months ago
Vocaloid:

that automatically eliminates the -2 - root(6) solution then I just plugged in -2 + root(6) into the original equation and the math worked out fine

4 months ago

The polynomial or the first ln equation

4 months ago
Vocaloid:

both should work

4 months ago
Vocaloid:

ln(-2 + sqrt(6) + 2) - ln(-2+sqrt(6)) = ln(-2 + sqrt(6) + 5)) if you are using a calculator be careful with parentheses

4 months ago
Vocaloid:

both sides are ~1.695521979

4 months ago