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Chapter 3: Problem 62

Write the exponential equation in logarithmic form. $$e^{1 / 3}=1.3956 \ldots$$

Short Answer

Expert verified
\(\log_e 1.3956 = \frac{1}{3}\)

Step by step solution

01

Understand the exponential form

The given formula is in the form: \(a^b = c\). Where base 'a' is the constant \(e\), 'b' is the exponent \(\frac{1}{3}\), and 'c' is the result \(1.3956.\)
02

Convert to logarithmic form

Convert by using the logarithmic definition. Logarithm base will be \(e\), the inside of logarithm will be \(1.3956\), and it will be equal to \(\frac{1}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Equation
An exponential equation involves a mathematical expression in which a constant base is raised to a variable exponent. Consider the equation \(e^{1/3} = 1.3956\). In this equation, we see that the base \(e\) is raised to a fractional power, \(\frac{1}{3}\), resulting in the value of approximately 1.3956.Here is a breakdown:
  • The base is the constant value that is raised to a power, in this case, \(e\).
  • The exponent is the power to which the base is raised. It is the \(\frac{1}{3}\) in our example.
  • The result or the answer of the exponential expression is the number obtained, which is 1.3956 here.
Exponential equations are widely used in science and finance, they model processes that grow or decay at exponential rates. Understanding their form is key to grasping the concept of exponential growth.
Base e
The base \(e\) is a special mathematical constant approximately equal to 2.71828. It appears often in mathematics, especially in calculus and complex analysis. This number is important because it is the natural base for logarithms, often used in continuous growth models.The following are some important things about \(e\):
  • It is also known as Euler's number, named after the mathematician Leonhard Euler.
  • \(e\) is fundamental in natural logarithms, as natural logarithms use \(e\) as their base.
  • It is an irrational number. This means it cannot be exactly expressed as a fraction.
  • Its application ranges from calculating compound interest to probability theory and even in the field of engineering.
Learning about \(e\) is crucial for students needing to solve exponential equations involving natural growth processes. Grasping its importance can aid in understanding more advanced topics later on.
Exponential Form
Exponential form is a way of representing numbers or expressions using a base raised to an exponent, such as \(a^b = c\). This format is useful in simplifying expressions and solving equations.For the given example \(e^{1/3} = 1.3956\):
  • The base is \(e\), a fundamental mathematical constant.
  • The exponent is the power assigned to the base, in this form, \(\frac{1}{3}\).
  • The result is the numeric value achieved when the base is computed with the given exponent, 1.3956 here.
When writing exponential equations in their exponential form, the relationship between the base, exponent, and result is clearly presented, making the mathematical properties easier to interpret. This allows for a bridge to transform it into logarithmic form when needed, providing flexibility in mathematical problem-solving.

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Most popular questions from this chapter

The graph of a Gaussian model is ________ shaped, where the ________ ________ is the maximum -value of the graph.

Find the exponential model \(y=a e^{b x}\) that fits the points shown in the graph or table. $$ \begin{array}{|l|l|l|} \hline x & 0 & 3 \\ \hline y & 1 & \frac{1}{4} \\ \hline \end{array} $$

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