Problem 258 - Calculus Volume 1

Question

For the following exercises, write the equation in equivalent logarithmic form. $$ \left(\frac{1}{3}\right)^{3}=\frac{1}{27} $$

Step-by-Step Solution

Answer

\( \log_{\frac{1}{3}} \frac{1}{27} = 3 \)

Step 1: Identify the Given Exponential Form

The equation \( \left(\frac{1}{3}\right)^{3}=\frac{1}{27} \) is given in exponential form. Here, the base is \( \frac{1}{3} \), the exponent is \( 3 \), and the result is \( \frac{1}{27} \).

Step 2: Structure of Logarithmic Form

Recall the structure for converting an exponential equation to logarithmic form: If \( a^b = c \), then in logarithmic form, it is \( \log_a c = b \).

Step 3: Apply Structure to Given Equation

Apply the structure identified in Step 2 to the given equation \( \left(rac{1}{3}\right)^{3}=\frac{1}{27} \). Here, \( a = \frac{1}{3} \), \( b = 3 \), and \( c = \frac{1}{27} \).

Step 4: Write the Equivalent Logarithmic Form

Convert the exponential equation to logarithmic form using the formula from Step 2: \[ \log_{\frac{1}{3}} \frac{1}{27} = 3 \]

Step 5: Verify the Conversion

Verify that our conversion is accurate. The expression \( \log_{\frac{1}{3}} \frac{1}{27} = 3 \) indicates that raising the base \( \frac{1}{3} \) to the power of 3 results in \( \frac{1}{27} \), which matches our original equation.

Key Concepts

Exponential Equation

An exponential equation is a mathematical expression in which a constant base is raised to a variable exponent. This type of equation is fundamental because it describes rapid growth or decay, and it's prevalent in various mathematical and real-world situations.
The given example, \( \left(\frac{1}{3}\right)^{3}=\frac{1}{27} \), is an exponential equation. Here, the base is \( \frac{1}{3} \), and it is raised to the power of 3, which is the exponent. The result obtained is \( \frac{1}{27} \).
Exponential equations can be represented in the general form \( a^b = c \), where:

• \( a \) is the base,
• \( b \) is the exponent,
• \( c \) is the result.

Understanding this form is crucial when dealing with any problem involving exponential growth, decay, or transformation into logarithmic form.

Base-Exponent-Result Relationship

The relationship between the base, exponent, and result is a cornerstone of exponential equations. This connection is also vital when converting exponential equations into logarithmic form.
Let's break it down with the equation \( \left(\frac{1}{3}\right)^{3}=\frac{1}{27} \):

• Base (\( \frac{1}{3} \)): The number that is repeatedly multiplied by itself.
• Exponent (3): Indicates how many times the base is multiplied.
• Result (\( \frac{1}{27} \)): The final product obtained after performing the exponential operation.

When these components interact, they create an equation that can be translated into logarithmic form, offering a different perspective on the same mathematical relationship.

Converting Logarithmic Expressions

Conversion from exponential to logarithmic form is a key skill in mathematics, making complex equations easier to handle. The process involves understanding the structure that connects these two concepts.
Any exponential equation \( a^b = c \) can be rewritten in logarithmic form as \( \log_a c = b \). This transformation highlights the exponent as a result of the logarithm of the base and the result.
For the equation \( \left(\frac{1}{3}\right)^{3}=\frac{1}{27} \), you can convert it to \( \log_{\frac{1}{3}} \frac{1}{27} = 3 \). This states that 3 is the power to which \( \frac{1}{3} \) must be raised to get \( \frac{1}{27} \).
Understanding how to switch between these forms is essential for solving equations and grasping more advanced mathematical concepts with ease.