# Natural log's and trigonometric identity's.

• stf
In summary, the student attempted to solve a problem but was not able to understand why ln 1 equals zero. They found another rule that states ln(a^b)=b*ln(a). Using this information they were able to solve for x.
stf

## Homework Statement

prove the identity.

ln csc(x) = -ln sin(x)

## The Attempt at a Solution

With a calculator I am able to prove it is in fact an identity, but I have NO clue why? In fact I am quite unclear on logarrithms in general as my algebra class just barely covered them and did not mention the natural log's at all except in passing (As "it would be covered in calculus")...So any help is great as to what's going on!

stf said:

## Homework Statement

prove the identity.

ln csc(x) = -ln sin(x)

## The Attempt at a Solution

With a calculator I am able to prove it is in fact an identity, but I have NO clue why? In fact I am quite unclear on logarrithms in general as my algebra class just barely covered them and did not mention the natural log's at all except in passing (As "it would be covered in calculus")...So any help is great as to what's going on!

ln means logarithm base e, where e is about 2.71828.

What properties of logarithms, in any base, do you know? One of them is the key to this problem, along with one trig identity.

Ok, Revisiting logarithm rules, it looks like it would be ln 1/sin = ln 1 - ln sinx, so I am assuming that ln 1 = 0 which gives the equation...I still don't fully understand why ln 1 equals zero though...I suppose I need to revisit all this old material. Thanks.

Yes, or equivalently, there's another rule that states $$ln(a^b)=b*ln(a)$$ and in this case $$csc(x)=(sin(x))^{-1}$$ so b=-1.

Log is the inverse of the exponential e, just like the square root is the inverse of squaring. If we have $$y=e^x$$ then $$x=ln(y)$$ such as if we have $$y=x^2$$ then $$x=\sqrt{y}$$.

So ln(1)=x, well this is the same as saying $$1=e^x$$ and for what value of x does $$e^x=1$$?

Mentallic said:
Yes, or equivalently, there's another rule that states $$ln(a^b)=b*ln(a)$$ and in this case $$csc(x)=(sin(x))^{-1}$$ so b=-1.

Log is the inverse of the exponential e, just like the square root is the inverse of squaring. If we have $$y=e^x$$ then $$x=ln(y)$$ such as if we have $$y=x^2$$ then $$x=\sqrt{y}$$.

So ln(1)=x, well this is the same as saying $$1=e^x$$ and for what value of x does $$e^x=1$$?

you forgot an important fact

$$e^{-ln(x)} = \frac{1}{x}$$

If he/she uses that fact then it explains the rest :)

I did? I showed the OP the two basic rules that he needs and can manipulate those in such a way to get the answer.

$$e^{-lnx}=\frac{1}{x}$$ isn't so quickly obvious if you're only just starting to learn the rules $$e^{lnx}=x$$ and $$ln(a^b)=b*lna$$

But you can easily get the result by using these rules that they are learning.

Mentallic said:
I did? I showed the OP the two basic rules that he needs and can manipulate those in such a way to get the answer.

$$e^{-lnx}=\frac{1}{x}$$ isn't so quickly obvious if you're only just starting to learn the rules $$e^{lnx}=x$$ and $$ln(a^b)=b*lna$$

But you can easily get the result by using these rules that they are learning.

Oh well, I am sure that Jedi Hal will find this post and correct us all, but that said

I learned in High School $$x^{-1} = \frac{1}{x}$$ which implies

$$e^{-x} = \frac{1}{e^x}$$ thus

$$e^{-ln(x)} = \frac{1}{e^{ln(x)}} = \frac{1}{x}$$

Use this property to prove you problem, OP!

Who's Jedi Hal?

Sure, there's lots of ways to do it.

$$e^{-ln(x)}=e^{ln(x^{-1})}=x^{-1}=\frac{1}{x}$$

$$e^{-ln(x)}=(e^{ln(x)})^{-1}=(x)^{-1}=\frac{1}{x}$$

$$e^{-ln(x)}=(e^{-1})^{ln(x)}=\left(\frac{1}{e}\right)^{ln(x)}=\frac{1}{e^{ln(x)}}=\frac{1}{x}$$

## 1. What is the definition of natural logarithm?

The natural logarithm is a mathematical function that is the inverse of the exponential function. It is denoted as ln(x) and represents the power to which the base number e (approximately 2.71828) must be raised to equal a given number x.

## 2. How is the natural logarithm related to exponential functions?

The natural logarithm and exponential functions are inverse operations of each other. This means that if we take the natural logarithm of an exponential function, we will get the exponent as the result. Similarly, if we take the exponential of a natural logarithm, we will get the original number.

## 3. What are some common properties of natural logarithms?

Some common properties of natural logarithms include: the natural logarithm of 1 is 0, the natural logarithm of e is 1, and the natural logarithm of a product is the sum of the natural logarithms of the individual factors.

## 4. What are the most commonly used trigonometric identities?

Some of the most commonly used trigonometric identities include: the Pythagorean identities (sin²x + cos²x = 1), the double angle identities (sin2x = 2sinxcosx), and the half angle identities (sin²(x/2) = (1-cosx)/2).

## 5. How are trigonometric identities used in mathematics and science?

Trigonometric identities are used in a variety of mathematical and scientific fields, such as geometry, calculus, physics, and engineering. They are used to simplify equations, solve problems, and model real-world phenomena such as waves, oscillations, and circular motion.

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