Question

Prove that \(\ln x^{y}=y \ln x\)

Short answer

The identity \(\ln x^{y}=y \ln x\) has been proved using the properties of logarithms and exponents.

Step-by-step solution

Identify the law to prove and analyze

We want to prove the logarithmic identity \(\ln x^{y}=y \ln x\). This form is often referred to as the power rule of logarithms. This rule is applicable when there is an exponent in a logarithmic function. The exponent y goes out in front and the function can be rewritten as multiplication.

Apply the definition of the natural logarithm

The natural logarithm \(\ln\) is the inverse of the exponential function with the base \(e\). This implies that if we have \(\ln x^{y}\), we can write this in exponential form as \(e^{\ln x^{y}}\). But this is the same as \(x^{y}\) because the exponential and the logarithm cancel out.

Apply the basic rule of exponents

After getting the exponential form in the last step, apply the basic rule of exponents: \(e^{a} \cdot e^{b}=e^{a+b}\). Now \(e^{y \ln x}\) can be rewritten as \(e^{\ln x^{y}}\).

Reverse transform to the logarithm form

After step 3, we know that both \(e^{y \ln x}\) and \(x^{y}\) are the same. By taking the natural logarithm of both sides, we obtain that \(y \ln x = \ln x^{y}\). Because the natural logarithm is a function, the rule that equal arguments yield equal function values applies.

Key concepts

Logarithmic Identity

Logarithmic identity is fundamental to the workings of logarithms and constitutes one of the first principles that students encounter when learning about this topic. The natural logarithm, denoted as (), is a special logarithm with the base equivalent to Euler's number, which is approximately 2.71828. A key identity associated with natural logarithms is the power rule, which states ( x^{y} ) = y ( x ).

This identity implies that you can take the exponent in the argument of the logarithm and move it to the front as a multiplier. Thus, this property makes it immensely easier to work with exponential expressions within logarithms. It is essential for students to understand that this happens because logarithms are essentially exponents themselves; they represent the power to which the base (in this case, e) must be raised to produce the given number.

To make this more consumable for students, here's an analogy: Imagine a box where you put the argument of the logarithm inside, and the exponent is like a label on the box. The logarithmic identity allows us to move the label and stick it onto the outside of the box, not changing the contents but altering how we describe what's inside. This mental model can help in visualizing the process of applying the logarithmic identity.

Exponential Functions

Exponential functions are a class of mathematical functions that are profoundly important in both theoretical and practical applications. The simplest exponential function is ( e^{x} ), where e is Euler's number. These functions describe situations where the rate of change is proportional to the value of the function itself—this is why they frequently appear in contexts such as compound interest, population growth, and radioactive decay.

Understanding exponential functions is crucial when dealing with natural logarithms because the natural logarithm is the inverse of the exponential function with base e. In other words, if you have an exponential function like ( e^{y} ) and you apply the natural logarithm to it, you will simply get y, which is the power to which e was raised. Think of exponential functions and natural logarithms as two sides of the same coin; they 'undo' each other. An analogy could be that if the exponential function is like climbing up a ladder, then the natural logarithm is like climbing back down to where you started. This relationship is pivotal in transforming complex logarithmic expressions into more manageable forms.

Rules of Exponents

The rules of exponents are a set of guidelines for performing operations on exponential expressions. These rules are not just arbitrary conventions but arise naturally from the way exponentiation is defined. Among these rules, a well-known one is the product rule: ( e^{a} ) ( e^{b} ) = e^{a+b} . The product rule is often encountered when multiplying two exponential expressions with the same base, dictating that you can add the exponents together.

Another key rule is the power rule, which states that ( (e^{a})^{b} ) = e^{ab} . This rule is useful when an exponent is raised to another exponent. Students should internalize these rules as they are not just algebraic tools; they reflect the inherent properties of numbers and operations.

Understanding the Product Rule

For instance, if you have two separate instances of growth, one growing at a rate of ( e^{a} ) and another at ( e^{b} ), when you combine them, you're not just putting these two rates side by side; you're actually compounding their effects. That's why their powers add up—to reflect the compounded growth rate. It's like two currents merging into a stronger single current. This conceptual understanding can help demystify why exponents behave the way they do and enhance the student's ability to work with them in a more intuitive manner.