Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the logarithmic equation in exponential form. For example, the exponential form of \(\ln 5=1.6094\). . . is \(e^{1.6094 \cdots}=5.\) $$\ln \sqrt[3]{e}=\frac{1}{3}$$

Step-by-Step Solution

Answer

The given logarithmic equation in exponential form is \(e^{1/3} = e^{1/3}\)

Step 1: Replace cube root with equivalent exponent

Express the cube root of e as \(e^{1/3}\). So the equation becomes \(\ln e^{1/3}=\frac{1}{3}\)

Step 2: Transform the logarithmic equation into an exponential equation

Since \(\ln\) represents the logarithm of base e, it can be written using exponents as \(e^{1/3} = e^{1/3}\)

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a special kind of logarithm with base e, where \(e\) is an irrational constant approximately equal to 2.71828. In mathematics, the natural logarithm is the inverse operation of taking \(e\) to the power of a number, symbolized as \(e^{x}\). For instance, if you have the natural logarithm of a number \(x\), expressed as \(\ln x\), and you want to convert it to exponential form, you'd write it as \(e^{\ln x} = x\).

The natural logarithm is widely used in many areas of mathematics and science because it simplifies many formulas and calculations, especially when dealing with continuous growth or decay, such as in the contexts of compound interest, population growth, or radioactive decay. An important property to remember is that \(\ln e = 1\), because \(e\) raised to the power of 1 is still \(e\). Our example exercise demonstrates this by taking the cube root of \(e\), which can be expressed as \(e^{1/3}\) and shows that the natural logarithm of \(e^{1/3}\) is \(1/3\).

Exponential Equations

Exponential equations are equations in which the variable appears in an exponent. They take the form of \(a^{x} = b\), where \(a\) and \(b\) are numbers, and \(x\) is the variable. To solve such equations, one often utilizes logarithms. The natural logarithm is especially useful for solving exponential equations with the base \(e\). For example, if you have the equation \(e^{x} = b\), taking the natural logarithm of both sides will simplify the equation to \(x = \ln b\).

Solving exponential equations requires a firm grasp of both logarithms and exponents. The solution to the exercise above involves recognizing that the logarithmic function \(\ln\) and the exponent to the base \(e\) are inverse operations. So, the equation \(\ln e^{1/3} = \frac{1}{3}\) can be interpreted using the property that \(\ln e^{x} = x\), giving us the exponential form of \(e^{1/3} = e^{\ln e^{1/3}} = e^{1/3}\), which shows that the equations are equivalent and thus correctly transformed.

Properties of Logarithms

Properties of logarithms are essential for understanding how to work with logarithmic expressions and solve logarithmic equations. Some basic properties include:

• The product rule: \(\ln(ab) = \ln a + \ln b\)
• The quotient rule: \(\ln(\frac{a}{b}) = \ln a - \ln b\)
• The power rule: \(\ln(a^{n}) = n\ln a\)

These properties are derived from the fundamental nature of logarithms as the inverse of exponentiation and can simplify complex logarithmic expressions. They can be thought of as 'tools' that allow us to deconstruct and reconstruct logarithmic expressions more conveniently.

Our exercise uses the power rule to simplify the logarithm of a power of \(e\). Since \(\ln(e^{1/3})\) equals \(1/3\times\ln e\) and knowing that \(\ln e = 1\), it simplifies directly to \(\frac{1}{3}\), illustrating the property's practical usage. It's also important for students to practice these properties to develop an intuition for log functions and be able to transition between logarithmic and exponential forms seamlessly.