I did this in a slightly different (worse?) way. I solved for x using the Lambert W function and got x = e^(-W(-4 c)/4). The W function isn't defined for arguments < -1/e. It is equal to -1 at -1/e, has two values between -1/e and 0, and has a single value when the argument is greater than zero. Since all of the answer options are positive, the argument to W will be negative - and so it must be -1/e, the only negative argument that yields a single real value, and therefore c must be 1/(4e).
Actually the LambertW function has 0 real solutions on the interval (-infinity; -1/e), 2 solutions on the interval [-1/e; 0) and has one solution on the interval [0; infinity) and the argument of the function is -4c, therefore, c is either a number form (-infinity; -1/e) or the argument has to be -1/e, we still get one solutions for this value, so we set -4c=-1/e and we get c=1/4e
Derivative if log(x) is not 1/x. Graphing tech confirms that the given solution does not work for this question. If you meant ln(x) = cx^4, then it does work. The way it is wtitten, the correct answer is c = log(e)/(4e)
It’s obvious from context here that it’s the natural log. It’s pretty much always the natural log when doing something purely mathematical, unless stated otherwise.
The answer is correct, but somewhere in the middle of the step is incorrect because the derivative of log(x) is 1/(xln(10)). From there, if you differentiate both sides of the equation, you get 1/(xln(10))=4cx^3, so 4cx^4ln(10)=1. Now substitute cx^4=log(x), which is 4log(x)ln(10)=1. Note that log(x)=ln(x)/ln(10). Basically, 4ln(x)=1, so x=e^(1/4). If you raise both sides by 4, we have x^4=e. Putting back in the equation that was differentiated, we 4ce=1. Solving for c gives us c=1/(4e). Hence, the answer must be choice A.
@@mathoutloudlog x usually means to the natural logarithm log x could also be to binary logarithm or base ten logarithm. Definitely not always and should be specified 😊
I did this in a slightly different (worse?) way. I solved for x using the Lambert W function and got x = e^(-W(-4 c)/4). The W function isn't defined for arguments < -1/e. It is equal to -1 at -1/e, has two values between -1/e and 0, and has a single value when the argument is greater than zero. Since all of the answer options are positive, the argument to W will be negative - and so it must be -1/e, the only negative argument that yields a single real value, and therefore c must be 1/(4e).
Actually the LambertW function has 0 real solutions on the interval (-infinity; -1/e), 2 solutions on the interval [-1/e; 0) and has one solution on the interval [0; infinity) and the argument of the function is -4c, therefore, c is either a number form (-infinity; -1/e) or the argument has to be -1/e, we still get one solutions for this value, so we set -4c=-1/e and we get c=1/4e
What a nice problem! Really puts into question if the examinee understands the meaning of a derivative!
Derivative if log(x) is not 1/x. Graphing tech confirms that the given solution does not work for this question. If you meant ln(x) = cx^4, then it does work. The way it is wtitten, the correct answer is c = log(e)/(4e)
Last time I checked that’s what it is for positive values of x.
Many times in math log is used to denote the natural logarithm or base e
You should specify whether it is natural log or not@@mathoutloud
It’s obvious from context here that it’s the natural log. It’s pretty much always the natural log when doing something purely mathematical, unless stated otherwise.
The answer is correct, but somewhere in the middle of the step is incorrect because the derivative of log(x) is 1/(xln(10)). From there, if you differentiate both sides of the equation, you get 1/(xln(10))=4cx^3, so 4cx^4ln(10)=1. Now substitute cx^4=log(x), which is 4log(x)ln(10)=1. Note that log(x)=ln(x)/ln(10). Basically, 4ln(x)=1, so x=e^(1/4). If you raise both sides by 4, we have x^4=e. Putting back in the equation that was differentiated, we 4ce=1. Solving for c gives us c=1/(4e). Hence, the answer must be choice A.
Here log is representing the natural logarithm and not the base 10 log very common in upper level mathematics
Differentiating does not always work
But in this case, simple sketches show u that it does
Please mention the base of the log in the question
It is always e.
@@mathoutloud That makes sense because I solved it using base of 10 cause you used log and not ln
@@prathamkalgutkar7538 yeah, depends on the field that you mainly studied in
maths often take base e for log, while physics often takes base 10
@@alonewanderer4697bro log is for log base 10, ln is for log base e
@@mathoutloudlog x usually means to the natural logarithm log x could also be to binary logarithm or base ten logarithm. Definitely not always and should be specified 😊
ez pz