how to solve $log (x+1) +3 = frac2x+2$

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From the Jan 17 NYS Algebra 2 Regents, a multiple choice question with seemingly no satisfying answer:



"When $g(x) = frac2x+2$ and $h(x) = log (x+1)+3$ are graphed on the same set of axes, which coordinates best approximate their point of intersection?"



$$(1) (-0.9, 1.8)$$
$$(2) (-0.9, 1.9)$$
$$(3) (1.4, 3.3)$$
$$(4) (1.4, 3.4)$$



Plugging in an $x$ value of $-0.9$ yields $y$ values of $1.8181$... and $2$, respectively, so neither answers # 1 nor # 2 are 'best,' and an $x$-value of $1.4$ does not work at all for $g(x)$.



On Desmos I obtained an approximation of $(-.927, 1.8638)$ as the intersection of the graphs, i.e., approximately the 'solution.'



Here's my real question: can this type of equation, $frac2x+2 = log(x+1)+3$ actually be solved algebraically; and if so, how?



Many thanks, as always.




logarithms




share|cite|improve this question

















  • 1




    Maybe you want to try the "Lambert W Function" for reference.
    – Evan William Chandra
    Jul 23 at 4:49










  • Only hope is numeric. "Lambert W Function" will not help you in any way.
    – Mariusz Iwaniuk
    Jul 23 at 8:19










  • Can't see a closed form solution but consider $log(x+1)-dfrac2x+2+3 = 0$ and apply newtons method to find $x$
    – Warren Hill
    Jul 23 at 14:38















up vote
1
down vote

favorite












From the Jan 17 NYS Algebra 2 Regents, a multiple choice question with seemingly no satisfying answer:



"When $g(x) = frac2x+2$ and $h(x) = log (x+1)+3$ are graphed on the same set of axes, which coordinates best approximate their point of intersection?"



$$(1) (-0.9, 1.8)$$
$$(2) (-0.9, 1.9)$$
$$(3) (1.4, 3.3)$$
$$(4) (1.4, 3.4)$$



Plugging in an $x$ value of $-0.9$ yields $y$ values of $1.8181$... and $2$, respectively, so neither answers # 1 nor # 2 are 'best,' and an $x$-value of $1.4$ does not work at all for $g(x)$.



On Desmos I obtained an approximation of $(-.927, 1.8638)$ as the intersection of the graphs, i.e., approximately the 'solution.'



Here's my real question: can this type of equation, $frac2x+2 = log(x+1)+3$ actually be solved algebraically; and if so, how?



Many thanks, as always.




logarithms




share|cite|improve this question

















  • 1




    Maybe you want to try the "Lambert W Function" for reference.
    – Evan William Chandra
    Jul 23 at 4:49










  • Only hope is numeric. "Lambert W Function" will not help you in any way.
    – Mariusz Iwaniuk
    Jul 23 at 8:19










  • Can't see a closed form solution but consider $log(x+1)-dfrac2x+2+3 = 0$ and apply newtons method to find $x$
    – Warren Hill
    Jul 23 at 14:38













up vote
1
down vote

favorite









up vote
1
down vote

favorite











From the Jan 17 NYS Algebra 2 Regents, a multiple choice question with seemingly no satisfying answer:



"When $g(x) = frac2x+2$ and $h(x) = log (x+1)+3$ are graphed on the same set of axes, which coordinates best approximate their point of intersection?"



$$(1) (-0.9, 1.8)$$
$$(2) (-0.9, 1.9)$$
$$(3) (1.4, 3.3)$$
$$(4) (1.4, 3.4)$$



Plugging in an $x$ value of $-0.9$ yields $y$ values of $1.8181$... and $2$, respectively, so neither answers # 1 nor # 2 are 'best,' and an $x$-value of $1.4$ does not work at all for $g(x)$.



On Desmos I obtained an approximation of $(-.927, 1.8638)$ as the intersection of the graphs, i.e., approximately the 'solution.'



Here's my real question: can this type of equation, $frac2x+2 = log(x+1)+3$ actually be solved algebraically; and if so, how?



Many thanks, as always.




logarithms




share|cite|improve this question













From the Jan 17 NYS Algebra 2 Regents, a multiple choice question with seemingly no satisfying answer:



"When $g(x) = frac2x+2$ and $h(x) = log (x+1)+3$ are graphed on the same set of axes, which coordinates best approximate their point of intersection?"



$$(1) (-0.9, 1.8)$$
$$(2) (-0.9, 1.9)$$
$$(3) (1.4, 3.3)$$
$$(4) (1.4, 3.4)$$



Plugging in an $x$ value of $-0.9$ yields $y$ values of $1.8181$... and $2$, respectively, so neither answers # 1 nor # 2 are 'best,' and an $x$-value of $1.4$ does not work at all for $g(x)$.



On Desmos I obtained an approximation of $(-.927, 1.8638)$ as the intersection of the graphs, i.e., approximately the 'solution.'



Here's my real question: can this type of equation, $frac2x+2 = log(x+1)+3$ actually be solved algebraically; and if so, how?



Many thanks, as always.





logarithms





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 23 at 4:30









Gonzalo Benavides

581317




581317









asked Jul 23 at 4:18









Victor Jaroslaw

212




212







  • 1




    Maybe you want to try the "Lambert W Function" for reference.
    – Evan William Chandra
    Jul 23 at 4:49










  • Only hope is numeric. "Lambert W Function" will not help you in any way.
    – Mariusz Iwaniuk
    Jul 23 at 8:19










  • Can't see a closed form solution but consider $log(x+1)-dfrac2x+2+3 = 0$ and apply newtons method to find $x$
    – Warren Hill
    Jul 23 at 14:38













  • 1




    Maybe you want to try the "Lambert W Function" for reference.
    – Evan William Chandra
    Jul 23 at 4:49










  • Only hope is numeric. "Lambert W Function" will not help you in any way.
    – Mariusz Iwaniuk
    Jul 23 at 8:19










  • Can't see a closed form solution but consider $log(x+1)-dfrac2x+2+3 = 0$ and apply newtons method to find $x$
    – Warren Hill
    Jul 23 at 14:38








1




1




Maybe you want to try the "Lambert W Function" for reference.
– Evan William Chandra
Jul 23 at 4:49




Maybe you want to try the "Lambert W Function" for reference.
– Evan William Chandra
Jul 23 at 4:49












Only hope is numeric. "Lambert W Function" will not help you in any way.
– Mariusz Iwaniuk
Jul 23 at 8:19




Only hope is numeric. "Lambert W Function" will not help you in any way.
– Mariusz Iwaniuk
Jul 23 at 8:19












Can't see a closed form solution but consider $log(x+1)-dfrac2x+2+3 = 0$ and apply newtons method to find $x$
– Warren Hill
Jul 23 at 14:38





Can't see a closed form solution but consider $log(x+1)-dfrac2x+2+3 = 0$ and apply newtons method to find $x$
– Warren Hill
Jul 23 at 14:38
















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