Question

solve the logarithmic equation: ln x +ln(x+1)=5

Answer 1

hi @chelle1018 :)
do you know the log property that
\(\ln A+\ln B = \ln (AB)\)
?

Answer 2

using that property , left side will equal
\(\ln [x(x+1)]\)
got this ?

Answer 3

so:
\[e^5=x(x+1)\]
or \[e^5=x^2+x\]

Answer 4

actually you can rewrite it as:\[\frac{ e^5 }{ x }-x=1\]

Answer 5

so:\[\frac{ e^5 }{ x }-x=e^0\]

Answer 6

x^2+x -e^5 =0
is quadratic equation, solve it using quadratic formula

Answer 7

multiply both sides by x:\[e^5-x^2=x\]

Answer 8

if you plug that quadratic equation into the quadratic formula you get
\[x=\frac{ -1-\sqrt{1+4e^5} }{ 2 }\]

Answer 9

thats the answer

Answer 10

or \[x=\frac{ \sqrt{1+4e^5}-1 }{ 2 }\]