Problem 73 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(e^{2}=7.3890 . . .\) is \(\ln 7.3890 . . .=2.\) $$\sqrt[3]{e}=1.3956 . . . $$

Step-by-Step Solution

Answer

The logarithmic form of the provided equation is \( 1/3 = \ln (1.3956... ) \)

Step 1: Identify exponential equation

The provided exponential equation is \( \sqrt[3]{e} = 1.3956... \)

Step 2: Write it as e raised to 1/3

Express \( \sqrt[3]{e} \) as \( e ^ {1/3} \). Hence, our equation becomes \( e^{1/3} = 1.3956... \)

Step 3: Convert to logarithmic form

Using the logarithm property mentioned in the analysis, we convert the above equation to logarithmic form, \( 1/3 = \ln (1.3956... ) \).

Key Concepts

Exponential Equation

An exponential equation is one in which a variable appears in the exponent and the base is a constant. For instance, the equation \( e^{x} = y \) represents an exponential relationship where \( e \) is the base and \( x \) is the exponent. Solving exponential equations usually requires finding the value of the exponent that makes the equation true. A common way to solve them is by taking the logarithm of both sides, which transforms the exponential equation into a linear one, making it easier to solve for the variable.

To show this process using the given exercise as an example, we start with the exponential equation \( \sqrt[3]{e} = 1.3956... \) which can also be written as \( e^{1/3} = 1.3956... \). The goal is to solve for the exponent, which is \( \frac{1}{3} \) in this case. Taking the natural logarithm on both sides would result in the equation \( \frac{1}{3} = \ln(1.3956...) \) as this isolates the exponent and makes it calculable. It's important to remember that understanding exponential equations is crucial as they appear frequently in various fields such as finance, physics, and biology.

Natural Logarithm (ln)

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \) where \( e \) is an irrational and transcendental number approximately equal to 2.71828. It is a special and widely used case of logarithms in mathematics because of its natural properties in relation to growth processes. For instance, the continuous growth of bacteria or the compounding of interest can be described using the natural logarithm.

The function \( \ln(x) \) is the inverse of the exponential function \( e^{x} \) which means if \( e^{x} = y \) then \( \ln(y) = x \). In the context of our exercise, \( e^{1/3} = 1.3956... \) becomes \( 1/3 = \ln(1.3956...) \) when we apply the natural logarithm. The significance of the natural logarithm cannot be overstated as it plays an integral part in calculus, aiding in the integration and differentiation of exponential functions.

Properties of Logarithms

The properties of logarithms are mathematical rules that simplify the computation and manipulation of logarithmic expressions. Some of the fundamental properties include the product rule, the quotient rule, and the power rule.

• The product rule states that the logarithm of a product is the sum of the logarithms: \( \ln(xy) = \ln(x) + \ln(y) \).
• The quotient rule describes how the logarithm of a quotient is the difference of the logarithms: \( \ln(\frac{x}{y}) = \ln(x) - \ln(y) \).
• The power rule allows us to bring the exponent down as a coefficient: \( \ln(x^{n}) = n \cdot \ln(x) \).

During the solution of our example exercise, the power rule is visibly in action as the cube root \( \sqrt[3]{e} \) is expressed as \( e^{1/3} \) and when converted to logarithmic form it becomes \( 1/3 = \ln(1.3956...) \). These properties streamline solving logarithmic equations and understanding exponential relationships, therefore, being highly beneficial to students across various mathematical contexts.