Problem 113 - College Algebra with Modeling and Visualization

Question

Change each equation to its equivalent logarithmic form. (a) \(7^{4 x}=4\) (b) \(e^{x}=7\) (c) \(c^{x}=b\)

Step-by-Step Solution

Answer

(a) \(4x = \log_{7}(4)\); (b) \(x = \ln(7)\); (c) \(x = \log_{c}(b)\).

Step 1: Understanding Exponential Equations

To convert an exponential equation to its equivalent logarithmic form, remember: if you have an equation of the form \( a^{b} = c \), the logarithmic form is \( b = \log_{a}(c) \). Let's apply this to each part of the given exercise.

Step 2: Convert Equation (a)

For part (a), the equation is \( 7^{4x} = 4 \). According to the rule, we convert this to logarithmic form as: \( 4x = \log_{7}(4) \).

Step 3: Convert Equation (b)

For part (b), the equation is \( e^{x} = 7 \). Here the base is \( e \), which is the natural logarithm base. Therefore, the logarithmic form of the equation is: \( x = \ln(7) \), where \( \ln \) represents the natural logarithm.

Step 4: Convert Equation (c)

For part (c), the equation is \( c^{x} = b \). Applying the conversion rule here, the logarithmic form of this equation is: \( x = \log_{c}(b) \).

Key Concepts

Exponential Equations

Exponential equations are mathematical statements that involve the use of exponents. They take the form \( a^b = c \), where \( a \) is the base, \( b \) is the exponent, and \( c \) is the result of raising \( a \) to the power of \( b \). Exponential equations are commonly used in various fields such as finance, physics, and biology because they can express large numbers and exponential growth or decay.Understanding these types of equations helps in solving real-world problems and is essential for moving into more advanced mathematical concepts. In the case of the given exercise:

• Equation (a) is \( 7^{4x} = 4 \).
• Equation (b) is \( e^x = 7 \).
• Equation (c) is \( c^x = b \).

Each of these equations can be rewritten in a different form, called the logarithmic form, which uses logarithms instead of exponents.

Logarithmic Form

Logarithms are the inverse operations of exponents. Converting an exponential equation to its logarithmic form can make it easier to work with, especially when solving for unknown variables. The connection between exponential equations and logarithms can be summarized as follows:If you have an exponential equation like \( a^b = c \), its logarithmic form is \( b = \log_a(c) \). This tells us how many times we need to multiply \( a \) by itself to get \( c \).For the exercise:

• Equation (a) \( 7^{4x} = 4 \) converts to \( 4x = \log_7(4) \).
• Equation (b) \( e^x = 7 \) converts to \( x = \ln(7) \).
• Equation (c) \( c^x = b \) converts to \( x = \log_c(b) \).

Using the logarithmic form is particularly useful in solving for the exponent, which is often a challenging step in problems involving exponents.

Natural Logarithm

The natural logarithm is a special type of logarithm with the base \( e \), where \( e \) is approximately equal to 2.71828. It is denoted as \( \ln \) and is particularly important in calculus and mathematical modeling due to its unique properties.In the context of the exercise, the equation \( e^x = 7 \) demonstrates the use of the natural logarithm. The equivalent logarithmic form is \( x = \ln(7) \), meaning we are finding the power to which \( e \) must be raised to produce 7. Natural logarithms simplify calculations involving exponential growth and decay, such as compound interest or population growth models. They are widely used because they make complex equations easier to solve and understand, especially as they relate to the rate of growth.