Question

Inverse hyperbolic tangent Recall that the inverse hyperbolic tangent is defined as \(y=\tanh ^{-1} x \Leftrightarrow x=\tanh y,\) for \(-1

Short answer

Question: Find the inverse hyperbolic tangent function in terms of logarithms. Answer: The inverse hyperbolic tangent function in terms of logarithms is given by the formula: $$\tanh^{-1} x = \ln\left(\frac{1+x}{1-x}\right).$$

Step-by-step solution

Substitute the definition of hyperbolic tangent

We know that \(\tanh y = \frac{\sinh y}{\cosh y}\). So let's substitute this definition into the equation \(x = \tanh y\): $$x = \frac{\sinh y}{\cosh y}.$$

Substitute the definitions of hyperbolic sine and cosine

Now, we need to replace \(\sinh y\) and \(\cosh y\) with their definitions: $$x = \frac{\frac{e^y - e^{-y}}{2}}{\frac{e^y + e^{-y}}{2}}.$$

Simplify the equation

We have a complex fraction here, so let's simplify it by multiplying the numerator and denominator by 2: $$x = \frac{e^y - e^{-y}}{e^y + e^{-y}}.$$

Prepare to isolate \(y\) from the equation

Our next goal is to isolate \(y\). Firstly, let's multiply both sides by the denominator, \(e^y + e^{-y}\): $$x(e^y + e^{-y}) = e^y - e^{-y}.$$ Now, distribute \(x\) on the left side and get rid of the negative exponent on the right side: $$xe^y + xe^{-y} = e^y - e^{-y}.$$

Make \(e^y\) the subject of the equation

To do this, we'll subtract \(xe^y\) and add \(e^{-y}\) to both sides of the equation: $$(1-x)e^y = (1+x)e^{-y}.$$ Now, we will isolate \(e^y\). Divide both sides by \((1-x)\): $$e^{y} = \frac{1+x}{1-x} e^{-y}.$$ To fully isolate \(e^y\), let's take the reciprocal of both sides: $$e^{-y} = \frac{1-x}{1+x} \implies e^{y} = \frac{1+x}{1-x}.$$

Convert to a logarithm

Now that we have \(e^y\) isolated, we can use a logarithm to find \(y\). Take the natural logarithm of both sides: $$y = \ln\left(\frac{1+x}{1-x}\right).$$ We have now found the formula for the inverse hyperbolic tangent function in terms of logarithms: $$\tanh^{-1} x = \ln\left(\frac{1+x}{1-x}\right).$$

Key concepts

Hyperbolic Functions

Hyperbolic functions are analogs to the trigonometric functions but for a hyperbola rather than a circle. They are important in various areas of mathematics, including algebra and calculus, and have applications in fields like physics and engineering.

The two most basic hyperbolic functions are the hyperbolic sine and cosine, denoted as \(\sinh x\) and \(\cosh x\), respectively. These functions are defined in terms of exponential functions: \[\sinh x = \frac{e^{x} - e^{-x}}{2}\] and \[\cosh x = \frac{e^{x} + e^{-x}}{2}.\]

One characteristic of hyperbolic functions that aligns with trigonometric functions is that they can be defined using the unit hyperbola \(x^2 - y^2 = 1\), much like trigonometric functions are related to the unit circle. However, unlike trigonometric functions, hyperbolic functions do not repeat their values in 'cycles' because they are not periodic.

The hyperbolic tangent function \(\tanh x\), for example, is given by the ratio \(\tanh x = \frac{\sinh x}{\cosh x}\) and has a range from -1 to 1. As we approach the values of -1 or 1, the hyperbolic tangent function grows steeply, which reflects the asymptotic behavior of the hyperbola.

Understanding these hyperbolic functions is crucial when working with the inverse hyperbolic tangent problem, as it sets the foundation for how we express the solution in terms of logarithms.

Natural Logarithm

The natural logarithm, denoted as \(\ln x\), is the inverse of the exponential function where the base is the special constant \(e\), approximately equal to 2.71828.

The natural logarithm has a domain of \(x > 0\) because you cannot take the logarithm of a negative number or zero in real numbers. It is a fundamental tool in calculus and appears frequently in formulas across mathematics and physics.

The relationship between logarithms and exponentiation is critical to understanding the process of solving for the inverse hyperbolic tangent. Specifically, \(\ln(e^x) = x\) and \(e^{\ln x} = x\). The unique property that \(\ln(e^x)\) simply equals \(x\) allows us to 'undo' the effect of an exponential function, enabling us to solve equations for the exponent—in this case, the variable \(y\) in the inverse hyperbolic tangent function. \[\tanh^{-1} x = \ln\left(\frac{1+x}{1-x}\right)\]

By transforming the expression for \(\tanh y\) using exponential definitions and then isolating the \(e^y\) term, we can apply the natural logarithm to solve for \(y\), revealing the inverse hyperbolic tangent.

Exponential Functions

Exponential functions are mathematical functions of the form \(f(x) = e^x\), where \(e\) is the constant approximately equal to 2.71828, also known as Euler's number.

One of the most important properties of the exponential function is its derivative, which is unique among all functions in that it is equal to itself: \(\frac{d}{dx}e^x = e^x\). This makes it incredibly useful in modeling growth processes that are self-similar at different scales, such as compound interest, population growth, or radioactive decay.

In the context of our inverse hyperbolic tangent problem, exponential functions allow us to express hyperbolic functions. When solving for \(y\) in the equation \(x = \tanh y\), we encounter exponential functions when expanding the hyperbolic sine and cosine.

The manipulations involving exponential functions finally lead us to an equation where we can isolate \(e^y\) and take its natural logarithm to find \(y\). This marriage of exponential functions and natural logarithms is what allows us to express the inverse hyperbolic tangent in terms of logarithms, as shown in the final step of the solution \[\tanh^{-1} x = \ln\left(\frac{1+x}{1-x}\right).\]

Therefore, understanding exponential functions is not only crucial for solving the problem at hand but also for appreciating the interplay between various mathematical concepts.