Question

Select the answer that completes the statement: \(y=\ln x\) if and only if _____. (a) \(x=e^{y}\) (b) \(y=e^{x}\) (c) \(x=10^{y}\) (d) \(y=10^{x}\)

Short answer

The correct answer is (a) \(x = e^y\).

Step-by-step solution

Identify the given equation

The given equation is: \(y = \ln x\).

Understand the definition of the natural logarithm

Recall that the natural logarithm \(\ln x\) is the inverse of the exponential function with base \(e\). This means that if \(y = \ln x\), then \(x\) can be expressed as an exponential function of \(y\).

Rewrite the equation in exponential form

Since \(y = \ln x\) is the natural logarithm, its inverse relationship is represented by the equation: \(x = e^y\).

Match the correct option

Compare the rewritten exponential form \(x = e^y\) with the given options: (a) \(x = e^y\) (b) \(y = e^x\) (c) \(x = 10^y\) (d) \(y = 10^x\). The option that matches is (a) \(x = e^y\).

Key concepts

Inverse Functions

Inverse functions are pairs of functions that 'undo' each other. When we apply an inverse function to the result of its corresponding function, we get back our original value. For example, if you have a function \(f(x)\) and its inverse is \(f^{-1}(x)\), then \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).
An important pair of inverse functions in mathematics are exponential functions and logarithmic functions. If we have an exponential function \(y = e^x\), its inverse is the natural logarithm function \(x = \,\ln y\). This relationship means that the natural logarithm of a number is the exponent to which you must raise \(e\) (approximately 2.718) to get that number.
Understanding inverse functions helps us solve equations where we need to 'undo' operations. In the original exercise, knowing that the natural logarithm \(\ln x\) and the exponential function \(e^y\) are inverses led us to the correct answer.

Exponential Functions

Exponential functions are mathematical functions where a constant base is raised to a variable exponent. The general form is \(f(x) = b^x\) where \(b\) is the base and \(x\) is the exponent. A common base for exponential functions is the mathematical constant \(e\) (approximately 2.718). When we use \(e\) as the base, the exponential function is written as \(f(x) = e^x\).
These functions grow rapidly; small changes in the exponent result in large changes in the output. Exponential functions are used in many fields including science, finance, and engineering to model growth processes like population growth, radioactive decay, and compound interest.
In the exercise, recognizing that the relationship between \(\ln x\) and \(e^y\) is that of an exponential function was crucial. It helps us see that if \(y = \ln x\), then \(x = e^y\).

Logarithmic Equations

Logarithmic equations involve logarithms, which are the inverses of exponential functions. The general form of a logarithmic equation is \(y = \log_b(x)\), where \(b\) is the base of the logarithm and \(x\) is the argument of the logarithm. The equation tells us the power \(y\) to which we must raise the base \(b\) to obtain \(x\).
Natural logarithms have a base of \(e\), and the equation \(y = \ln x\) means that \(e^y = x\). This transformation is essential for solving logarithmic equations. For instance, to solve \(\ln x = y\), we rewrite it in its exponential form \(x = e^y\).
In our original exercise, solving for \(y = \ln x\) required us to use the properties of logarithmic equations. We converted the logarithmic form to its exponential equivalent to find that the corresponding relationship is \(x = e^y\). This insight allowed us to identify the correct answer, knowing that the natural logarithm is the inverse of the exponential function with base \(e\).