How Do You Isolate x in This Logarithmic Trigonometric Equation?

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Homework Help Overview

The discussion revolves around isolating the variable x in the logarithmic trigonometric equation 20*log10(√(x² + 1)/cos(x)) = 3. Participants are exploring the implications of logarithmic properties and trigonometric identities in their attempts to manipulate the equation.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants discuss dividing the equation and applying logarithmic properties, such as log10(u/v) = log10(u) - log10(v). There are attempts to isolate cos(x) and questions about how to handle the logarithmic component. Some participants express confusion about the validity of certain equalities and the implications of using numerical methods for solving the equation.

Discussion Status

The discussion is ongoing, with various approaches being explored. Some participants have suggested using trigonometric identities and numerical methods, while others are questioning the assumptions and interpretations of the equations presented. There is no explicit consensus on the best path forward, but several productive lines of reasoning have been initiated.

Contextual Notes

Participants note the importance of correctly applying logarithmic properties and the potential need for numerical methods due to the complexity of isolating x. There is also mention of the challenges posed by the trigonometric functions involved in the equation.

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Homework Statement



Im supposed to calculate the value of x for

20*log10(\frac{\sqrt{x^2 +1}}{cos(x)}) = 3




The Attempt at a Solution


I start off by dividing by 20 on both sides so I just have the expression with log.. and my idea henceforth is to take the inverse of cosine to get rid of cos(x), and square the square root to cancel out.. but the logarithm part has me stuck.. and I am not sure how to get rid of/move it.. I've thought about multiplying with its inverse, but that has not gotten me any results.. any tips?
 
Last edited:
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How about e?

But I can't really tell what to do after that.
 
Tyrone123 said:

Homework Statement



I'm supposed to calculate the value of x for

20*log\frac{\sqrt{x^2 +1}}{\cos(x)}

The Attempt at a Solution


I start off by dividing by 20 on both sides so I just have the expression with log.. and my idea henceforth is to take the inverse of cosine to get rid of cos(x), and square the square root to cancel out.. but the logarithm part has me stuck.. and I'm not sure how to get rid of/move it.. I've thought about multiplying with its inverse, but that has not gotten me any results.. any tips?

Homework Statement


Homework Equations


The Attempt at a Solution

You edited your original post!

Now, there is no equation!

(Formerly it said 20*log\frac{\sqrt{x^2 +1}}{\cos(x)}=3\ .)
 
SammyS said:
You edited your original post!

Now, there is no equation!

(Formerly it said 20*log\frac{\sqrt{x^2 +1}}{\cos(x)}=3\ .)

woops, I edited log to log10.. and suddenly the equation was gone :b
 
I think propertieds of logs are important here.

log10 u/v = log10(u)-log10(v)

So, your original equation can be rewritten as:
20(log10(u)-log10(v))=3
with the numerator and denominator substituted for u and v, respectively. (hopefuly that makes sense)

Does that help?
 
ohms law said:
I think propertieds of logs are important here.

log10 u/v = log10(u)-log10(v)

So, your original equation can be rewritten as:
20(log10(u)-log10(v))=3
with the numerator and denominator substituted for u and v, respectively. (hopefuly that makes sense)

Does that help?

Possibly, ill try
 
Not really.. I end up with a slightly prettier equation, but I am not sure how to isolate x from log(cos(x)
 
ah, well, there's a trig identity to help with that, I'm sure...
Off of the top of my head, there's always cos^2(x)+sin^2(x)=1, so...
I'll have to look some stuff up and see if it'll help.
 
If \log_{10}(A)=B\,, then A=10^B\ .
 
  • #10
All I see right now is something like this (where I'm using 10 instead of e) \sqrt{x^2 + 1} = a \cos{x} where a = 10^{3/20} then use the trig identity ohms law mentioned. (But multiply the identity by -1)
 
  • #11
ohms law said:
ah, well, there's a trig identity to help with that, I'm sure...
Off of the top of my head, there's always cos^2(x)+sin^2(x)=1, so...
I'll have to look some stuff up and see if it'll help.

Hmm.. so far I have at least gotten rid of the log part, and I now have:

\frac{\sqrt{x^2 +1}}{cos(x)} = 10\frac{3}{20}

Which I can rearrange a bit.. but that cos(x) is causing me troubles!
 
  • #12
fyi, for the numerator portion, I simplified it to 1/3 log(x)
 
  • #13
Tyrone123 said:
Hmm.. so far I have at least gotten rid of the log part, and I now have:

\frac{\sqrt{x^2 +1}}{\cos(x)} = 10\frac{3}{20}

Which I can rearrange a bit.. but that cos(x) is causing me troubles!

I do believe that solving this requires using a numerical method.
 
  • #14
1/cos(x) would = sec(x)...
so, log(1/cos(x)) = log(1) - log(sec(x))
since log(1) = 0, we can just get rid of that

just not sure how much that helps
 
  • #15
SammyS said:
I do believe that solving this requires using a numerical method.

so, you're saying just plug and puke?
:)
 
  • #16
ohms law said:
so, you're saying just plug and puke?
:)
I've reviewed my posts, and no, I didn't say that. (LOL !)

OP may want to put his equation into a more suitable form before trying a numerical method.

Also, a graph may help in choosing a starting approximation.
 
Last edited:
  • #17
Tyrone123 said:
Hmm.. so far I have at least gotten rid of the log part, and I now have:

\frac{\sqrt{x^2 +1}}{cos(x)} = 10\frac{3}{20}

Which I can rearrange a bit.. but that cos(x) is causing me troubles!

I missed this earlier, but... I'm not sure what you mean with that equality. It's not true as stated (see graph):
Isolating a function variable graph.jpg
 
  • #18
SammyS said:
I do believe that solving this requires using a numerical method.

I guess that I'm just not clear what you mean by "requires using a numerical method".
I feel like I should know what you mean, but... :blushing:
 
  • #19
ohms law said:
I missed this earlier, but... I'm not sure what you mean with that equality. It's not true as stated (see graph):
View attachment 51351

I don't mean anything in particular, just me trying to isolate x in the equation
 
  • #20
Tyrone123 said:
Hmm.. so far I have at least gotten rid of the log part, and I now have:

\frac{\sqrt{x^2 +1}}{cos(x)} = 10\frac{3}{20}

Which I can rearrange a bit.. but that cos(x) is causing me troubles!

ohms law said:
I missed this earlier, but... I'm not sure what you mean with that equality. It's not true as stated (see graph):
View attachment 51351
Actually, that equality is true !

It leads to \displaystyle \sqrt{x^2+1}=10^{3/20}\cos(x)\ .

Tyrone123,
Notice that \left(10^{3/20}\right)^2\approx1.99526\,, which is very nearly 2, so that 10^{3/20} is just a little less that \sqrt{2}\ .

Also, it may help to square that above equation, which then becomes:
\displaystyle x^2 +1=10^{3/10}\cos^2(x)\ .​

For small values of x, a good approximation for cos(x) is:
\displaystyle <br /> \cos(x)\approx 1-\frac{x^2}{2}\ .​

Plug that into the previous equation and solve for x to get a good starting place for any numerical method you choose.
 
  • #21
SammyS said:
Actually, that equality is true !

It leads to \displaystyle \sqrt{x^2+1}=10^{3/20}\cos(x)\ .

man, trig makes my head hurt...
I don't get it. I'd drop it, but my sense is that this is part of the problem that Tyrone is having too (maybe I'm wrong, but...).

I know that graphs don't prove anything, but the graphs are (extremely) different (even with the cos(x) product added to the one side), so... I'm confused.
 
  • #22
Here are the graphs of y=\sqrt{x^2+1} and y=10^{3/20}\cos(x) overlaid on the same graph (from WolframAlpha).

attachment.php?attachmentid=51362&stc=1&d=1349039177.gif
 

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  • #23
ohms law said:
I missed this earlier, but... I'm not sure what you mean with that equality. It's not true as stated (see graph):
View attachment 51351
What is graphed here?

attachment.php?attachmentid=51351&d=1349029962.jpg
 
  • #24
ohms law said:
man, trig makes my head hurt...
I don't get it. I'd drop it, but my sense is that this is part of the problem that Tyrone is having too (maybe I'm wrong, but...).

I know that graphs don't prove anything, but the graphs are (extremely) different (even with the cos(x) product added to the one side), so... I'm confused.
Take Tyrone's original equation.
\displaystyle 20\,\log_{10}\left(\frac{\sqrt{x^2 +1}}{\cos(x)}\right)=3\​

Do a little basic algebra & apply some of the log rules you suggested:
\displaystyle \log(\sqrt{x^2+1})-\log(\cos(x))=\frac{3}{20}

\displaystyle \log(\sqrt{x^2+1})=\frac{3}{20}+\log(\cos(x))
​

If you use each side as the exponent with a base of 10, you get:
\displaystyle \sqrt{x^2+1}=10^{3/20}\cos(x)\​
 

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