Problem 7 - A Graphical Approach to Precalculus with Limits

Question

For each statement, write an equivalent statement in logarithmic form. Do not use a calculator. $$e^{0}=1$$

Step-by-Step Solution

Answer

\( \log_{e}(1) = 0 \)

Step 1: Identify the Given Exponential Form

The given statement is in exponential form: \( e^{0} = 1 \). Here, the base is \( e \), the exponent is \( 0 \), and the result is \( 1 \).

Step 2: Recall the Logarithmic Form

The equivalent logarithmic form of an exponential statement \( a^{b} = c \) is \( \log_{a}(c) = b \).

Step 3: Substitute into Logarithmic Form

Using the logarithmic form \( \log_{a}(c) = b \), substitute \( a = e \), \( b = 0 \), and \( c = 1 \) into the formula to get the logarithmic form: \( \log_{e}(1) = 0 \).

Key Concepts

Exponential Functions

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. This type of function is often represented as \( a^x \), where \( a \) is the base and \( x \) is the exponent.Exponential functions are important in fields such as biology, finance, and physics due to their ability to model rapid growth or decay.

• The base \( a \) must be a positive number.
• If \( a = e \), the function is considered a natural exponential function. Here, \( e \) is approximately equal to 2.71828...
• When \( x = 0 \), any non-zero base raised to the power of zero equals 1. Thus, \( e^{0} = 1 \).

Such characteristics make exponential functions flexible for many applications, ranging from calculating compound interest to evaluating population dynamics.

Equivalent Statements

In mathematics, expressing an idea through different but logically identical forms is known as equivalent statements.This approach is fundamental for solving equations and making sense of complex expressions.

• Equivalent statements convey the same information, although they are structured differently.
• Converting between forms, such as exponential and logarithmic, is a key skill in algebra.
• This practice helps in simplifying equations for easier manipulation and interpretation.

For instance, the equation \( e^{0} = 1 \) can be transformed while still retaining its inherent truth.Understanding this concept aids in solving equations quickly and enhances mathematical fluency.

Logarithmic Form

The logarithmic form of an equation helps express exponentiation in terms of logarithms, which are the inverses of exponential functions.A logarithm asks, "To what power must the base be raised to produce a given number?"In mathematical terms, if you have an exponential form \( a^{b} = c \), the logarithmic form would be \( \log_{a}(c) = b \).

• The base \( a \) of the exponential corresponds to the base of the logarithm.
• The result \( c \) of the exponential function becomes the number we are taking the logarithm of.
• The exponent \( b \) turns into the result of the logarithmic expression.

Using our example: for \( e^{0} = 1 \), the equivalent logarithmic form is \( \log_{e}(1) = 0 \). This transformation is useful as it allows one to switch between analyzing growth processes to solving for unknown exponents in scientific applications.