Problem 17 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite each equation in logarithmic form. \(c^{d}=k\)

Step-by-Step Solution

Answer

The logarithmic form of \( c^d = k \) is \( d = \log_c k \).

Step 1: Identify the Exponential Equation

We begin with the exponential equation given: \( c^d = k \). Our task is to express this equation in logarithmic form.

Step 2: Understand the Logarithmic Form

The general form for converting from exponential to logarithmic form is \( b^y = x \) converting to \( y = \log_b x \). Here, \( b \) is the base, \( y \) is the exponent, and \( x \) is the result.

Step 3: Apply the Conversion

In the equation \( c^d = k \), the base \( b \) is \( c \), the exponent \( y \) is \( d \), and the result \( x \) is \( k \). Therefore, applying the conversion, we get \( d = \log_c k \).

Step 4: Write the Logarithmic Form

Finally, the logarithmic form of the exponential equation \( c^d = k \) is \( d = \log_c k \).

Key Concepts

Exponential Equations

When discussing exponential equations, we are talking about equations in which a variable appears in the exponent. They follow the form \( b^y = x \), where \( b \) is the base, \( y \) is the exponent, and \( x \) is the result. This representation is quite prevalent in mathematical analysis and real-world applications such as compound interest, population growth, and radioactive decay.

• If you know the base and the result, solving for the exponent is straightforward once you learn the conversion to logarithmic form.
• Exponential equations can appear daunting, but they follow systematic rules that simplify the process of solving them.

It’s crucial to recognize when an expression or equation is exponential to effectively work with or manipulate it into forms we can manage. This foundational understanding helps in the conversion into logarithmic form, which is incredibly useful when our goal is to solve for the exponent.

Logarithmic Conversion

Logarithmic conversion is a pivotal concept in algebra, serving as the bridge between exponential and linear relationships. Converting an exponential equation to a logarithmic form involves using the formula \( b^y = x \) and rewriting it as \( y = \log_b x \). This process isn't just a mathematical trick; it provides a powerful tool for simplifying complex problems.

• Logarithms essentially answer the question: "To what power must this base be raised to produce this result?"
• They turn multiplicative processes into additive ones, making them easier to analyze and solve.

In the context of our exercise, the conversion of \( c^d = k \) to \( d = \log_c k \) allows us to express the understanding that \( c \) raised to the power \( d \) equals \( k \). This makes the process of finding "\( d \)" more straightforward, particularly when tackling problems manually or without a calculator.

Base and Exponent Concepts

Understanding the roles of base and exponent is fundamental when working with both exponential and logarithmic equations. In any expression \( b^y = x \), \( b \) represents the base, while \( y \) is the exponent. The base tells us what number is being repeatedly multiplied, and the exponent indicates how many times this multiplication happens.

• A larger exponent means the base is multiplied more times, which can grow the result rapidly if the base is greater than one.
• The base being less than one leads to a decay, meaning with increasing exponents, the outcome diminishes.

In logarithmic form, such as \( y = \log_b x \), the base remains the same as in the exponential form, emphasizing its consistency across conversions. Grasping these concepts enhances comprehension of exponential growth and decay, equipping students with a deeper understanding of mathematical dynamics in equations across multiple fields.