Problem 17 - Precalculus Mathematics for Calculus

Question

Express the equation in logarithmic form. (a) \(e^{x}=2\) (b) \(e^{3}=y\)

Step-by-Step Solution

Answer

(a) \( x = \ln(2) \); (b) \( 3 = \ln(y) \).

Step 1: Understand Exponential Form

The expression given is in the form of an exponential equation, which generally looks like \( a^b = c \), where \( a \) is the base, \( b \) is the exponent, and \( c \) is the result.

Step 2: Convert to Logarithmic Form

To convert an exponential equation \( a^b = c \) into its logarithmic form, use the conversion formula \( \log_a(c) = b \). This formula means that "\( b \) is the power to which the base \( a \) must be raised to obtain \( c \)."

Step 3: Apply Conversion to (a)

For (a) \( e^{x}=2 \), identify the base \( e \), the exponent \( x \), and the result \( 2 \). Using the conversion, we write it as a logarithm: \( x = \log_e(2) \) or \( x = \ln(2) \), because the natural logarithm is the logarithm to the base \( e \).

Step 4: Apply Conversion to (b)

For (b) \( e^{3}=y \), identify the base \( e \), the exponent \( 3 \), and the result \( y \). Using the conversion, we write it as a logarithm: \( 3 = \log_e(y) \) or \( 3 = \ln(y) \).

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a constant base is raised to a variable exponent, like the examples you see in our original exercise. These equations take the general form \( a^b = c \). Here, \( a \) is a fixed number known as the base, \( b \) is the exponent or the power, and \( c \) is the result. Exponential equations are notable for describing growth and decay processes, such as population growth, radioactive decay, and more.

• In the equation \( e^x = 2 \), \( e \) is the base, a special mathematical constant approximately equal to 2.718.
• The exponent \( x \) determines the power to which the base is raised.
• The result here is 2, indicating the output of the exponential function.

Understanding this basic form of an exponential equation is crucial for correctly communicating it in logarithmic form, which brings us to our next topic.

Natural Logarithms

Natural logarithms are a special type of logarithm where the base is the constant \( e \). When we say "natural logarithm," we denote it using \( \ln \). Natural logarithms are omnipresent in science and engineering, partly because they simplify many mathematical formulas and computations.

• The natural logarithm \( \ln(c) \) means "what power must \( e \) be raised to in order to get \( c \)."
• For example, \( e^x = 2 \) turns into \( x = \ln(2) \), meaning that \( x \) is the power needed to raise \( e \) to obtain 2.
• Similarly, if \( e^3 = y \), then \( 3 = \ln(y) \), suggesting \( y \) is the result when \( e \) is raised to the third power.

Natural logarithms are helpful due to their analytical properties, notably making the work of differentiation and integration more straightforward.

Logarithmic Conversion

Logarithmic conversion is the process of transforming exponential equations into their equivalent logarithmic form. This conversion allows us to work in a linear context, making it easier to solve equations where variables are exponents.
To convert from exponential to logarithmic form, remember the rule \( a^b = c \) becomes \( \log_a(c) = b \). This tells us that \( b \) is the exponent of \( a \) that results in \( c \). For natural logarithms, where the base is \( e \), the conversion involves \( \ln(c) = b \).

• Take \( e^x = 2 \). Here, converting gives us \( x = \ln(2) \).
• For \( e^3 = y \), the conversion results in \( 3 = \ln(y) \).

This conversion changes the perspective from viewing "how much" a quantity has grown (exponentially) to "how many times" (logarithmically) a multiplicative process, which is often more intuitive to understand and analyze.