Problem 21 - Contemporary Precalculus

Question

Translate the given exponential statement into an equivalent logarithmic statement. $$e^{12 / 7}=5.5527$$

Step-by-Step Solution

Answer

Based on the given exponential statement, $$e^{12/7} = 5.5527$$, convert it into its equivalent logarithmic statement. Answer: $$\ln{5.5527} = \frac{12}{7}$$.

Step 1: Identify Base, Exponent, and Result

In our given exponential statement, $$e^{12/7} = 5.5527$$, the base is e, the exponent is $${12 \over 7}$$, and the resulting value is 5.5527.

Step 2: Write the equivalent logarithmic statement

Using the relationship between exponential and logarithmic statements, we can rewrite our statement as $$\log_e{5.5527} = \frac{12}{7}$$. Notice that the base of the logarithm is e, which is also written as "ln" (natural logarithm). Therefore, we can rewrite the statement as: $$\ln{5.5527} = \frac{12}{7}$$.

Key Concepts

Exponential Functions

An exponential function is a mathematical function of the form \( f(x) = a^x \) where \( a \) is a constant, often referred to as the base of the function. In this function, \( a \) must be a positive number and not equal to 1. Exponential functions are characterized by their constant growth rate, meaning the function's growth doubles when \( x \) is increased by a constant number. This is because the variable, \( x \), is in the exponent position rather than being a base.

For instance, in the equation \( e^x \), \( e \) (approximately 2.718) is the base. This specific base is particularly important because \( e \) is the base of the natural logarithm and frequently appears in mathematics, especially in continuous growth processes such as compound interest and population growth. Exponential functions are the inverse of logarithmic functions, which means understanding one helps in understanding the other.

In our Original Exercise, \( e^{12/7} = 5.5527 \), you should notice that \( e \) is the base, and 12/7 is the exponent, making this an exponential function.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm with base \( e \). The natural logarithm of a number \( x \) is defined as the power to which \( e \) must be raised to result in \( x \). In mathematical terms, if \( \ln(x)=y \), then \( e^y=x \).
Furthermore, the natural logarithm is a powerful mathematical tool because of its simplicity in differentiating and integrating calculus problems. The relationship between exponential and natural logarithms makes it possible to solve many equations involving exponentials. For example, rewriting exponentials in their logarithmic form converts complex multiplicative processes into simpler additive ones.

In our example, \( \ln{5.5527} = \frac{12}{7} \), illustrates how an exponential equation can be rewritten as a logarithmic statement using the natural logarithm. Here, it means that \( e \) raised to the power of \( \frac{12}{7} \) gives 5.5527.

Logarithm Properties

Logarithm properties are fundamental rules that make it easier to work with logarithmic equations. These properties assist in solving equations, simplifying complex expressions, and transforming exponential forms into logarithmic forms.

Here are some essential logarithm properties:

• Product Rule: \( \log_b(xy) = \log_b(x) + \log_b(y) \)
• Quotient Rule: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \)
• Power Rule: \( \log_b(x^y) = y \cdot \log_b(x) \)
• Change of Base Formula: \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \) (for any positive \( k \))

Logarithm properties allow us to manipulate logarithmic expressions effectively, turning complex problems into more manageable forms. Understanding these properties can significantly aid in solving logarithmic and exponential equations, as seen in transforming the exponential statement \( e^{12/7} = 5.5527 \) to its logarithmic form \( \ln{5.5527} = \frac{12}{7} \). The natural logarithm's illustrative example demonstrates these transformations. By understanding these concepts, students can enhance their ability and confidence in solving a wide range of mathematical problems.