Mixing
$log_{0.1}(x^4) - 4 geq 0$ - Solution verification
$begingroup$
My question: are these steps ok?
To be more precise, is the step where I take fourth root ok?
$$log_{0.1}(x^4) - 4 geq 0$$
$$iff$$
$$log_{0.1}(x^4) geq 4$$
$$iff$$
$$log_{0.1}(x^4) geq log_{0.1}left(frac{1}{10000}right)$$
$$iff$$
$$x^4 leq frac{1}{10000}$$
$$iff$$
$$|x|leq frac{1}{10}$$
This is the solution without $x=0$.
proof-verification logarithms absolute-value
edited Jan 17 at 12:53
Robert Z
96.9k1066137
asked Jan 12 at 8:47
josfjosf
284217
$endgroup$
add a comment |
$begingroup$
My question: are these steps ok?
To be more precise, is the step where I take fourth root ok?
$$log_{0.1}(x^4) - 4 geq 0$$
$$iff$$
$$log_{0.1}(x^4) geq 4$$
$$iff$$
$$log_{0.1}(x^4) geq log_{0.1}left(frac{1}{10000}right)$$
$$iff$$
$$x^4 leq frac{1}{10000}$$
$$iff$$
$$|x|leq frac{1}{10}$$
This is the solution without $x=0$.
proof-verification logarithms absolute-value
edited Jan 17 at 12:53
Robert Z
96.9k1066137
asked Jan 12 at 8:47
josfjosf
284217
$endgroup$
$begingroup$
Sounds right to me...
$endgroup$
– Mostafa Ayaz
Jan 12 at 8:50
add a comment |
$begingroup$
My question: are these steps ok?
To be more precise, is the step where I take fourth root ok?
$$log_{0.1}(x^4) - 4 geq 0$$
$$iff$$
$$log_{0.1}(x^4) geq 4$$
$$iff$$
$$log_{0.1}(x^4) geq log_{0.1}left(frac{1}{10000}right)$$
$$iff$$
$$x^4 leq frac{1}{10000}$$
$$iff$$
$$|x|leq frac{1}{10}$$
This is the solution without $x=0$.
proof-verification logarithms absolute-value
edited Jan 17 at 12:53
Robert Z
96.9k1066137
asked Jan 12 at 8:47
josfjosf
284217
$endgroup$
My question: are these steps ok?
To be more precise, is the step where I take fourth root ok?
$$log_{0.1}(x^4) - 4 geq 0$$
$$iff$$
$$log_{0.1}(x^4) geq 4$$
$$iff$$
$$log_{0.1}(x^4) geq log_{0.1}left(frac{1}{10000}right)$$
$$iff$$
$$x^4 leq frac{1}{10000}$$
$$iff$$
$$|x|leq frac{1}{10}$$
This is the solution without $x=0$.
proof-verification logarithms absolute-value
proof-verification logarithms absolute-value
edited Jan 17 at 12:53
Robert Z
96.9k1066137
asked Jan 12 at 8:47
josfjosf
284217
edited Jan 17 at 12:53
Robert Z
96.9k1066137
asked Jan 12 at 8:47
josfjosf
284217
edited Jan 17 at 12:53
Robert Z
96.9k1066137
edited Jan 17 at 12:53
Robert Z
96.9k1066137
edited Jan 17 at 12:53
Robert Z
96.9k1066137
96.9k1066137
asked Jan 12 at 8:47
josfjosf
284217
asked Jan 12 at 8:47
josfjosf
284217
asked Jan 12 at 8:47
josfjosf
284217
284217
$begingroup$
Sounds right to me...
$endgroup$
– Mostafa Ayaz
Jan 12 at 8:50
add a comment |
$begingroup$
Sounds right to me...
$endgroup$
– Mostafa Ayaz
Jan 12 at 8:50
$begingroup$
Sounds right to me...
$endgroup$
– Mostafa Ayaz
Jan 12 at 8:50
$begingroup$
Sounds right to me...
$endgroup$
– Mostafa Ayaz
Jan 12 at 8:50
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
It is correct, but it would be easier to take into account that $log_{0.1}(x^4)=4log_{0.1}(x)$.
answered Jan 12 at 8:57
José Carlos SantosJosé Carlos Santos
160k22127232
$endgroup$
$begingroup$
Hmm...only true if $x>0$ of course.
$endgroup$
– drhab
Jan 12 at 9:00
$begingroup$
$vert xvert$ would be correct.
$endgroup$
– KM101
Jan 12 at 9:03
$begingroup$
@drhab RIght, but it is clear from the question that the OP is aware of that.
$endgroup$
– José Carlos Santos
Jan 12 at 9:05
add a comment |
$begingroup$
It’s completely fine, and you also correctly used the absolute value when taking the fourth root, but you can solve the inequality more quickly by noting:
$$log_a b^c = clog_a vert bvert tag{1}$$
$$log_{a^b} c = frac{1}{b}log_a c tag{2}$$
So, using $0.1 = 10^{-1}$, you can rewrite $log_{0.1} left(x^4right) geq 4$ as follows:
$$log_{0.1} left(x^4right) geq 4 iff -4log_{10} vert xvert geq 4 iff log_{10} vert xvert leq -1 iff vert xvert leq frac{1}{10}$$
Of course, as mentioned, your way is completely fine as well, but perhaps this way is a bit faster.
answered Jan 12 at 9:06
KM101KM101
5,9251524
$endgroup$
add a comment |
$begingroup$
For fun, an option :
$z:= log_{0.1}(x^4) ge 4;$
$x^4= (0.1)^z=(10^{-1})^z= (10)^{-z}$, i.e.
$(10)^z=x^{-4} ge (10)^4,$
finally $x le 1/(10)$.
Your answer is fine as has been pointed out.
answered Jan 12 at 10:50
Peter SzilasPeter Szilas
11.2k2822
$endgroup$
add a comment |
Your Answer
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
It is correct, but it would be easier to take into account that $log_{0.1}(x^4)=4log_{0.1}(x)$.
answered Jan 12 at 8:57
José Carlos SantosJosé Carlos Santos
160k22127232
$endgroup$
$begingroup$
Hmm...only true if $x>0$ of course.
$endgroup$
– drhab
Jan 12 at 9:00
$begingroup$
$vert xvert$ would be correct.
$endgroup$
– KM101
Jan 12 at 9:03
$begingroup$
@drhab RIght, but it is clear from the question that the OP is aware of that.
$endgroup$
– José Carlos Santos
Jan 12 at 9:05
add a comment |
$begingroup$
It is correct, but it would be easier to take into account that $log_{0.1}(x^4)=4log_{0.1}(x)$.
answered Jan 12 at 8:57
José Carlos SantosJosé Carlos Santos
160k22127232
$endgroup$
$begingroup$
Hmm...only true if $x>0$ of course.
$endgroup$
– drhab
Jan 12 at 9:00
$begingroup$
$vert xvert$ would be correct.
$endgroup$
– KM101
Jan 12 at 9:03
$begingroup$
@drhab RIght, but it is clear from the question that the OP is aware of that.
$endgroup$
– José Carlos Santos
Jan 12 at 9:05
add a comment |
$begingroup$
It is correct, but it would be easier to take into account that $log_{0.1}(x^4)=4log_{0.1}(x)$.
answered Jan 12 at 8:57
José Carlos SantosJosé Carlos Santos
160k22127232
$endgroup$
It is correct, but it would be easier to take into account that $log_{0.1}(x^4)=4log_{0.1}(x)$.
answered Jan 12 at 8:57
José Carlos SantosJosé Carlos Santos
160k22127232
answered Jan 12 at 8:57
José Carlos SantosJosé Carlos Santos
160k22127232
answered Jan 12 at 8:57
José Carlos SantosJosé Carlos Santos
160k22127232
answered Jan 12 at 8:57
José Carlos SantosJosé Carlos Santos
160k22127232
160k22127232
$begingroup$
Hmm...only true if $x>0$ of course.
$endgroup$
– drhab
Jan 12 at 9:00
$begingroup$
$vert xvert$ would be correct.
$endgroup$
– KM101
Jan 12 at 9:03
$begingroup$
@drhab RIght, but it is clear from the question that the OP is aware of that.
$endgroup$
– José Carlos Santos
Jan 12 at 9:05
add a comment |
$begingroup$
Hmm...only true if $x>0$ of course.
$endgroup$
– drhab
Jan 12 at 9:00
$begingroup$
$vert xvert$ would be correct.
$endgroup$
– KM101
Jan 12 at 9:03
$begingroup$
@drhab RIght, but it is clear from the question that the OP is aware of that.
$endgroup$
– José Carlos Santos
Jan 12 at 9:05
$begingroup$
Hmm...only true if $x>0$ of course.
$endgroup$
– drhab
Jan 12 at 9:00
$begingroup$
Hmm...only true if $x>0$ of course.
$endgroup$
– drhab
Jan 12 at 9:00
$begingroup$
$vert xvert$ would be correct.
$endgroup$
– KM101
Jan 12 at 9:03
$begingroup$
$vert xvert$ would be correct.
$endgroup$
– KM101
Jan 12 at 9:03
$begingroup$
@drhab RIght, but it is clear from the question that the OP is aware of that.
$endgroup$
– José Carlos Santos
Jan 12 at 9:05
$begingroup$
@drhab RIght, but it is clear from the question that the OP is aware of that.
$endgroup$
– José Carlos Santos
Jan 12 at 9:05
add a comment |
$begingroup$
It’s completely fine, and you also correctly used the absolute value when taking the fourth root, but you can solve the inequality more quickly by noting:
$$log_a b^c = clog_a vert bvert tag{1}$$
$$log_{a^b} c = frac{1}{b}log_a c tag{2}$$
So, using $0.1 = 10^{-1}$, you can rewrite $log_{0.1} left(x^4right) geq 4$ as follows:
$$log_{0.1} left(x^4right) geq 4 iff -4log_{10} vert xvert geq 4 iff log_{10} vert xvert leq -1 iff vert xvert leq frac{1}{10}$$
Of course, as mentioned, your way is completely fine as well, but perhaps this way is a bit faster.
answered Jan 12 at 9:06
KM101KM101
5,9251524
$endgroup$
add a comment |
$begingroup$
It’s completely fine, and you also correctly used the absolute value when taking the fourth root, but you can solve the inequality more quickly by noting:
$$log_a b^c = clog_a vert bvert tag{1}$$
$$log_{a^b} c = frac{1}{b}log_a c tag{2}$$
So, using $0.1 = 10^{-1}$, you can rewrite $log_{0.1} left(x^4right) geq 4$ as follows:
$$log_{0.1} left(x^4right) geq 4 iff -4log_{10} vert xvert geq 4 iff log_{10} vert xvert leq -1 iff vert xvert leq frac{1}{10}$$
Of course, as mentioned, your way is completely fine as well, but perhaps this way is a bit faster.
answered Jan 12 at 9:06
KM101KM101
5,9251524
$endgroup$
add a comment |
$begingroup$
It’s completely fine, and you also correctly used the absolute value when taking the fourth root, but you can solve the inequality more quickly by noting:
$$log_a b^c = clog_a vert bvert tag{1}$$
$$log_{a^b} c = frac{1}{b}log_a c tag{2}$$
So, using $0.1 = 10^{-1}$, you can rewrite $log_{0.1} left(x^4right) geq 4$ as follows:
$$log_{0.1} left(x^4right) geq 4 iff -4log_{10} vert xvert geq 4 iff log_{10} vert xvert leq -1 iff vert xvert leq frac{1}{10}$$
Of course, as mentioned, your way is completely fine as well, but perhaps this way is a bit faster.
answered Jan 12 at 9:06
KM101KM101
5,9251524
$endgroup$
It’s completely fine, and you also correctly used the absolute value when taking the fourth root, but you can solve the inequality more quickly by noting:
$$log_a b^c = clog_a vert bvert tag{1}$$
$$log_{a^b} c = frac{1}{b}log_a c tag{2}$$
So, using $0.1 = 10^{-1}$, you can rewrite $log_{0.1} left(x^4right) geq 4$ as follows:
$$log_{0.1} left(x^4right) geq 4 iff -4log_{10} vert xvert geq 4 iff log_{10} vert xvert leq -1 iff vert xvert leq frac{1}{10}$$
Of course, as mentioned, your way is completely fine as well, but perhaps this way is a bit faster.
answered Jan 12 at 9:06
KM101KM101
5,9251524
answered Jan 12 at 9:06
KM101KM101
5,9251524
answered Jan 12 at 9:06
KM101KM101
5,9251524
answered Jan 12 at 9:06
KM101KM101
5,9251524
5,9251524
add a comment |
add a comment |
$begingroup$
For fun, an option :
$z:= log_{0.1}(x^4) ge 4;$
$x^4= (0.1)^z=(10^{-1})^z= (10)^{-z}$, i.e.
$(10)^z=x^{-4} ge (10)^4,$
finally $x le 1/(10)$.
Your answer is fine as has been pointed out.
answered Jan 12 at 10:50
Peter SzilasPeter Szilas
11.2k2822
$endgroup$
add a comment |
$begingroup$
For fun, an option :
$z:= log_{0.1}(x^4) ge 4;$
$x^4= (0.1)^z=(10^{-1})^z= (10)^{-z}$, i.e.
$(10)^z=x^{-4} ge (10)^4,$
finally $x le 1/(10)$.
Your answer is fine as has been pointed out.
answered Jan 12 at 10:50
Peter SzilasPeter Szilas
11.2k2822
$endgroup$
add a comment |
$begingroup$
For fun, an option :
$z:= log_{0.1}(x^4) ge 4;$
$x^4= (0.1)^z=(10^{-1})^z= (10)^{-z}$, i.e.
$(10)^z=x^{-4} ge (10)^4,$
finally $x le 1/(10)$.
Your answer is fine as has been pointed out.
answered Jan 12 at 10:50
Peter SzilasPeter Szilas
11.2k2822
$endgroup$
For fun, an option :
$z:= log_{0.1}(x^4) ge 4;$
$x^4= (0.1)^z=(10^{-1})^z= (10)^{-z}$, i.e.
$(10)^z=x^{-4} ge (10)^4,$
finally $x le 1/(10)$.
Your answer is fine as has been pointed out.
answered Jan 12 at 10:50
Peter SzilasPeter Szilas
11.2k2822
answered Jan 12 at 10:50
Peter SzilasPeter Szilas
11.2k2822
answered Jan 12 at 10:50
Peter SzilasPeter Szilas
11.2k2822
answered Jan 12 at 10:50
Peter SzilasPeter Szilas
11.2k2822
11.2k2822
add a comment |
add a comment |
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$begingroup$
Sounds right to me...
$endgroup$
– Mostafa Ayaz
Jan 12 at 8:50