Problem 8 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, write the logarithmic equation as an exponential equation, or vice versa. $$ e^{0.25}=1.2840 . $$

Step-by-Step Solution

Answer

The logarithmic form of the equation \(e^{0.25} = 1.2840\) is \(ln 1.2840 = 0.25\).

Step 1: Understanding the exponential equation

Observe the equation \(e^{0.25} = 1.2840\). Here, 'e' is the base, 0.25 is the power or exponent, and 1.2840 is the result of the exponentiation.

Step 2: Applying the conversion formula

Now rewrite the equation in logarithmic form using the conversion formula. According to the formula \(\log_b q = p\), this translates to \(\log_e 1.2840 = 0.25\).

Step 3: Final simplification

The logarithm to the base 'e' is also known as the natural logarithm and it is denoted as ln. So, our final equation will be in the form \(ln 1.2840 = 0.25\).

Key Concepts

Exponential Equations

Exponential equations are fundamental in various fields of mathematics and science, as they describe situations where a quantity grows or decays at a rate proportional to its current value. An exponential equation takes the form \( b^x = y \) where \( b \) is the base, \( x \) is the exponent, and \( y \) is the result. In such equations, \( b \) is a constant and \( x \) is the variable.

The equation in our exercise, \( e^{0.25} = 1.2840 \), is a simple exponential equation where \( e \) is the base, known as Euler's number, approximately equal to 2.718. Here, \( e^{0.25} \) represents exponential growth at a certain rate, and the result of this growth is 1.2840.

Exponential functions are vital in modeling real world phenomena such as population growth, compound interest, radioactive decay, and much more. Understanding how to manipulate and solve these equations is a critical skill in mathematics.

Natural Logarithm

The natural logarithm is a type of logarithm with the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828, also known as Euler's number. In mathematical notation, the natural logarithm is represented as \( \ln(x) \) and it is the inverse operation of raising \( e \) to a power x.

For instance, if we have \( e^x = y \), the natural logarithm of \( y \) would be \( \ln(y) = x \). The natural logarithm is widely used because of its natural properties in calculus, particularly in differentiation and integration, where calculating the natural logarithm of a number simplifies various equations.

In the context of our exercise, converting \( e^{0.25} = 1.2840 \) to a logarithmic equation results in \( \ln 1.2840 = 0.25 \), demonstrating how to express an exponential equation involving the constant \( e \) into a logarithmic form.

Logarithm Properties

Logarithms have several properties that make them useful for simplifying complex algebraic expressions and solving equations. Some essential properties include:

• Product Rule: \( \log_b(xy) = \log_b(x) + \log_b(y) \), which states that the logarithm of a product is equal to the sum of the logarithms of the factors.
• Quotient Rule: \( \log_b(\frac{x}{y}) = \log_b(x) - \log_b(y) \), which tells us that the logarithm of a quotient is the difference of the logarithms.
• Power Rule: \( \log_b(x^p) = p\log_b(x) \), meaning that the logarithm of a number raised to a power is the power multiplied by the logarithm of the number.
• Change of Base Formula: \( \log_b(x) = \frac{\log_a(x)}{\log_a(b)} \), allowing us to change the base of a logarithm by rewriting it in terms of logarithms with a different base \( a \).

These properties are not merely theoretical; they are practical tools for solving logarithmic equations and understanding the behavior of logarithmic functions. For example, these properties allow you to simplify an expression like \( \log_e(1.2840) = 0.25 \) to \( \ln 1.2840 = 0.25 \), by recognizing the base \( e \) logarithm as the natural logarithm.