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Solve $100n^2 = 2^n$
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I don't understand how to solve this... I try to get rid of the $2^n$ via logarithms and it leads me to $2log_210 + 2log_2n = n$ and I don't know how to proceed.
Thanks for the help.
algebra-precalculus logarithms
edited Jul 22 at 18:23
Dan
25517
asked Jul 22 at 18:23
Shane
92
add a comment |Â
up vote
1
down vote
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I don't understand how to solve this... I try to get rid of the $2^n$ via logarithms and it leads me to $2log_210 + 2log_2n = n$ and I don't know how to proceed.
Thanks for the help.
algebra-precalculus logarithms
edited Jul 22 at 18:23
Dan
25517
asked Jul 22 at 18:23
Shane
92
This looks like an implicit equation to me or to put it in other words there is no explicit solution for this problem.
– mrtaurho
Jul 22 at 18:25
1
To solve this "explicitly" you need the Lambert W function. Numerically, the solutions are $n=0.1036578164, 14.32472784, -0.09670403432$
– GEdgar
Jul 22 at 18:27
Any advice on what to study next so I can better identify these and/or solve them in the future?
– Shane
Jul 22 at 18:28
1
I'd like to add that one can quickly see that no natural number $n$ solves the equation, since the left hand side then contains at least a couple of factors $5$, while the right hand side only contains factors $2$.
– md2perpe
Jul 22 at 18:38
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I don't understand how to solve this... I try to get rid of the $2^n$ via logarithms and it leads me to $2log_210 + 2log_2n = n$ and I don't know how to proceed.
Thanks for the help.
algebra-precalculus logarithms
edited Jul 22 at 18:23
Dan
25517
asked Jul 22 at 18:23
Shane
92
I don't understand how to solve this... I try to get rid of the $2^n$ via logarithms and it leads me to $2log_210 + 2log_2n = n$ and I don't know how to proceed.
Thanks for the help.
algebra-precalculus logarithms
edited Jul 22 at 18:23
Dan
25517
asked Jul 22 at 18:23
Shane
92
edited Jul 22 at 18:23
Dan
25517
edited Jul 22 at 18:23
Dan
25517
edited Jul 22 at 18:23
Dan
25517
25517
asked Jul 22 at 18:23
Shane
92
asked Jul 22 at 18:23
Shane
92
asked Jul 22 at 18:23
Shane
92
92
This looks like an implicit equation to me or to put it in other words there is no explicit solution for this problem.
– mrtaurho
Jul 22 at 18:25
1
To solve this "explicitly" you need the Lambert W function. Numerically, the solutions are $n=0.1036578164, 14.32472784, -0.09670403432$
– GEdgar
Jul 22 at 18:27
Any advice on what to study next so I can better identify these and/or solve them in the future?
– Shane
Jul 22 at 18:28
1
I'd like to add that one can quickly see that no natural number $n$ solves the equation, since the left hand side then contains at least a couple of factors $5$, while the right hand side only contains factors $2$.
– md2perpe
Jul 22 at 18:38
add a comment |Â
This looks like an implicit equation to me or to put it in other words there is no explicit solution for this problem.
– mrtaurho
Jul 22 at 18:25
1
To solve this "explicitly" you need the Lambert W function. Numerically, the solutions are $n=0.1036578164, 14.32472784, -0.09670403432$
– GEdgar
Jul 22 at 18:27
Any advice on what to study next so I can better identify these and/or solve them in the future?
– Shane
Jul 22 at 18:28
1
I'd like to add that one can quickly see that no natural number $n$ solves the equation, since the left hand side then contains at least a couple of factors $5$, while the right hand side only contains factors $2$.
– md2perpe
Jul 22 at 18:38
This looks like an implicit equation to me or to put it in other words there is no explicit solution for this problem.
– mrtaurho
Jul 22 at 18:25
This looks like an implicit equation to me or to put it in other words there is no explicit solution for this problem.
– mrtaurho
Jul 22 at 18:25
1
1
To solve this "explicitly" you need the Lambert W function. Numerically, the solutions are $n=0.1036578164, 14.32472784, -0.09670403432$
– GEdgar
Jul 22 at 18:27
To solve this "explicitly" you need the Lambert W function. Numerically, the solutions are $n=0.1036578164, 14.32472784, -0.09670403432$
– GEdgar
Jul 22 at 18:27
Any advice on what to study next so I can better identify these and/or solve them in the future?
– Shane
Jul 22 at 18:28
Any advice on what to study next so I can better identify these and/or solve them in the future?
– Shane
Jul 22 at 18:28
1
1
I'd like to add that one can quickly see that no natural number $n$ solves the equation, since the left hand side then contains at least a couple of factors $5$, while the right hand side only contains factors $2$.
– md2perpe
Jul 22 at 18:38
I'd like to add that one can quickly see that no natural number $n$ solves the equation, since the left hand side then contains at least a couple of factors $5$, while the right hand side only contains factors $2$.
– md2perpe
Jul 22 at 18:38
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
Lambert W method: $y = xe^x$ if and only if $W(y) = x$.
For this problem:
$$
100 n^2 = 2^n
$$
take square-root,
$$
10 n = 2^n/2qquadtextorqquad 10 n = -2^n/2
$$
Lets do the first one
$$
10 n = 2^n/2
\
10 n = expleft(fracn (log 2)2right)
\
frac110 n = expleft(frac-n (log 2)2right)
\
frac110 = n;expleft(frac-n (log 2)2right)
\
-fraclog 220 = frac-n log 22expleft(frac-n (log 2)2right)
$$
All of that was to get it in the form $y = x e^x$. Then go to $W(y) = x$:
$$
Wleft(-fraclog 220right) = frac-n (log 2)2
\
-frac2log 2Wleft(-fraclog 220right) = n
$$
Note. Students of high-school algebra are not expected to know the Lambert W funcion. And they are not expected to be able to solve this problem explicitly.
answered Jul 22 at 18:37
GEdgar
58.4k264163
Can you point me toward a math course or book that commonly covers this?
– Shane
Jul 22 at 19:34
@Shane: I've never seen the Lambert W function in a book, but it is covered in Wikipedia and in Wolfram MathWorld.
– Rory Daulton
Jul 23 at 0:10
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
Lambert W method: $y = xe^x$ if and only if $W(y) = x$.
For this problem:
$$
100 n^2 = 2^n
$$
take square-root,
$$
10 n = 2^n/2qquadtextorqquad 10 n = -2^n/2
$$
Lets do the first one
$$
10 n = 2^n/2
\
10 n = expleft(fracn (log 2)2right)
\
frac110 n = expleft(frac-n (log 2)2right)
\
frac110 = n;expleft(frac-n (log 2)2right)
\
-fraclog 220 = frac-n log 22expleft(frac-n (log 2)2right)
$$
All of that was to get it in the form $y = x e^x$. Then go to $W(y) = x$:
$$
Wleft(-fraclog 220right) = frac-n (log 2)2
\
-frac2log 2Wleft(-fraclog 220right) = n
$$
Note. Students of high-school algebra are not expected to know the Lambert W funcion. And they are not expected to be able to solve this problem explicitly.
answered Jul 22 at 18:37
GEdgar
58.4k264163
Can you point me toward a math course or book that commonly covers this?
– Shane
Jul 22 at 19:34
@Shane: I've never seen the Lambert W function in a book, but it is covered in Wikipedia and in Wolfram MathWorld.
– Rory Daulton
Jul 23 at 0:10
add a comment |Â
up vote
2
down vote
Lambert W method: $y = xe^x$ if and only if $W(y) = x$.
For this problem:
$$
100 n^2 = 2^n
$$
take square-root,
$$
10 n = 2^n/2qquadtextorqquad 10 n = -2^n/2
$$
Lets do the first one
$$
10 n = 2^n/2
\
10 n = expleft(fracn (log 2)2right)
\
frac110 n = expleft(frac-n (log 2)2right)
\
frac110 = n;expleft(frac-n (log 2)2right)
\
-fraclog 220 = frac-n log 22expleft(frac-n (log 2)2right)
$$
All of that was to get it in the form $y = x e^x$. Then go to $W(y) = x$:
$$
Wleft(-fraclog 220right) = frac-n (log 2)2
\
-frac2log 2Wleft(-fraclog 220right) = n
$$
Note. Students of high-school algebra are not expected to know the Lambert W funcion. And they are not expected to be able to solve this problem explicitly.
answered Jul 22 at 18:37
GEdgar
58.4k264163
Can you point me toward a math course or book that commonly covers this?
– Shane
Jul 22 at 19:34
@Shane: I've never seen the Lambert W function in a book, but it is covered in Wikipedia and in Wolfram MathWorld.
– Rory Daulton
Jul 23 at 0:10
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Lambert W method: $y = xe^x$ if and only if $W(y) = x$.
For this problem:
$$
100 n^2 = 2^n
$$
take square-root,
$$
10 n = 2^n/2qquadtextorqquad 10 n = -2^n/2
$$
Lets do the first one
$$
10 n = 2^n/2
\
10 n = expleft(fracn (log 2)2right)
\
frac110 n = expleft(frac-n (log 2)2right)
\
frac110 = n;expleft(frac-n (log 2)2right)
\
-fraclog 220 = frac-n log 22expleft(frac-n (log 2)2right)
$$
All of that was to get it in the form $y = x e^x$. Then go to $W(y) = x$:
$$
Wleft(-fraclog 220right) = frac-n (log 2)2
\
-frac2log 2Wleft(-fraclog 220right) = n
$$
Note. Students of high-school algebra are not expected to know the Lambert W funcion. And they are not expected to be able to solve this problem explicitly.
answered Jul 22 at 18:37
GEdgar
58.4k264163
Lambert W method: $y = xe^x$ if and only if $W(y) = x$.
For this problem:
$$
100 n^2 = 2^n
$$
take square-root,
$$
10 n = 2^n/2qquadtextorqquad 10 n = -2^n/2
$$
Lets do the first one
$$
10 n = 2^n/2
\
10 n = expleft(fracn (log 2)2right)
\
frac110 n = expleft(frac-n (log 2)2right)
\
frac110 = n;expleft(frac-n (log 2)2right)
\
-fraclog 220 = frac-n log 22expleft(frac-n (log 2)2right)
$$
All of that was to get it in the form $y = x e^x$. Then go to $W(y) = x$:
$$
Wleft(-fraclog 220right) = frac-n (log 2)2
\
-frac2log 2Wleft(-fraclog 220right) = n
$$
Note. Students of high-school algebra are not expected to know the Lambert W funcion. And they are not expected to be able to solve this problem explicitly.
answered Jul 22 at 18:37
GEdgar
58.4k264163
edited Jul 22 at 18:44
answered Jul 22 at 18:37
GEdgar
58.4k264163
answered Jul 22 at 18:37
GEdgar
58.4k264163
answered Jul 22 at 18:37
GEdgar
58.4k264163
58.4k264163
Can you point me toward a math course or book that commonly covers this?
– Shane
Jul 22 at 19:34
@Shane: I've never seen the Lambert W function in a book, but it is covered in Wikipedia and in Wolfram MathWorld.
– Rory Daulton
Jul 23 at 0:10
add a comment |Â
Can you point me toward a math course or book that commonly covers this?
– Shane
Jul 22 at 19:34
@Shane: I've never seen the Lambert W function in a book, but it is covered in Wikipedia and in Wolfram MathWorld.
– Rory Daulton
Jul 23 at 0:10
Can you point me toward a math course or book that commonly covers this?
– Shane
Jul 22 at 19:34
Can you point me toward a math course or book that commonly covers this?
– Shane
Jul 22 at 19:34
@Shane: I've never seen the Lambert W function in a book, but it is covered in Wikipedia and in Wolfram MathWorld.
– Rory Daulton
Jul 23 at 0:10
@Shane: I've never seen the Lambert W function in a book, but it is covered in Wikipedia and in Wolfram MathWorld.
– Rory Daulton
Jul 23 at 0:10
add a comment |Â
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This looks like an implicit equation to me or to put it in other words there is no explicit solution for this problem.
– mrtaurho
Jul 22 at 18:25
1
To solve this "explicitly" you need the Lambert W function. Numerically, the solutions are $n=0.1036578164, 14.32472784, -0.09670403432$
– GEdgar
Jul 22 at 18:27
Any advice on what to study next so I can better identify these and/or solve them in the future?
– Shane
Jul 22 at 18:28
1
I'd like to add that one can quickly see that no natural number $n$ solves the equation, since the left hand side then contains at least a couple of factors $5$, while the right hand side only contains factors $2$.
– md2perpe
Jul 22 at 18:38