Solving equation with natural log with inverse hyperbolic

In summary, the conversation discusses solving the equation ln(sinh-1(x)) = 1 without the use of a calculator. The formula for sinh-1(x) is given as asinh, but the equation becomes ln(asinh(x)) = 1, which may seem confusing. The solution involves taking the exponential of both sides, resulting in sinh-1(x) = e. Then, taking the sinh of both sides gives sinh(sinh-1(x)) = sinh(e), and using the inverse property of sinh, the equation simplifies to x = e.
  • #1
yojo95
5
0

Homework Statement



Solve ln(sinh-1(x)) = 1
(Just give the formula for x, no calculator is necessary)
sinh-1(x) = asinh


Homework Equations





The Attempt at a Solution



When I did this, I knew sinh-1(x) = ln(x + sqrt.(x^2 +1), but the equation would look like
ln(ln(x + sqrt.(x^2 + 1 )) = 1, which looks weird because I don't what to do afterwards
So then I tried exponentiate both sides and it will be asinh(x) = e then multiplying both sides by sinh, so my answer is x = sinhe, is this right?
 
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  • #2
You do NOT 'multiply both sides by sinh'- there is no multiplication involved here! After you take the exponential of both sides you have sinh-1(x)= e. Now take sinh of both sides of that: sinh(sinh-1(x))= sinh(e).

What is sinh(sinh-1(x))?
 
  • #3
oh sorry, What i meant was "take" not "multiply" even though i typed multiply -_-

ok so then, sinh(sinh-1(x)) = sinh-1(sinh(x)) = x ?
 

FAQ: Solving equation with natural log with inverse hyperbolic

1. What is the natural log with inverse hyperbolic?

The natural log with inverse hyperbolic is a mathematical function that is used to solve equations involving logarithmic and exponential functions. It is the inverse of the hyperbolic function, which is defined as the area under a hyperbola.

2. How do you solve equations with natural log and inverse hyperbolic?

To solve an equation with natural log and inverse hyperbolic, you must first isolate the logarithmic or exponential term on one side of the equation. Then, you can use the properties of logarithms and inverse hyperbolic functions to simplify the equation and find the solution.

3. What are the properties of natural log with inverse hyperbolic?

The properties of natural log with inverse hyperbolic include:

  • The natural log with inverse hyperbolic is the inverse of the hyperbolic function.
  • The natural log with inverse hyperbolic is defined as ln(x) = y if and only if ey = x.
  • The natural log with inverse hyperbolic has a base of e, the natural logarithm constant.
  • The natural log with inverse hyperbolic has a domain of all positive real numbers and a range of all real numbers.

4. When do you use natural log with inverse hyperbolic in equations?

Natural log with inverse hyperbolic is commonly used in equations that involve logarithmic or exponential functions. It is also used in solving equations involving hyperbolic functions, as it is the inverse of the hyperbolic function.

5. Are there any common mistakes when solving equations with natural log and inverse hyperbolic?

One common mistake when solving equations with natural log and inverse hyperbolic is forgetting to check the domain of the function. Since the natural log with inverse hyperbolic has a domain of all positive real numbers, any solutions that result in negative numbers should be discarded. Another mistake is not properly applying the properties of logarithms and inverse hyperbolic functions, which can lead to incorrect solutions.

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