Problem 31 - Algebra 2

Question

Write each equation in logarithmic form. \(5^{-3}=\frac{1}{125}\)

Step-by-Step Solution

Answer

\(\log_{5}\left(\frac{1}{125}\right) = -3\)

Step 1: Understand the Relationship

To convert the exponential equation to a logarithmic equation, we must identify the base of the exponent, the exponent itself, and the result. The given exponential equation is \(5^{-3} = \frac{1}{125}\). Here, the base is 5, the exponent is -3, and the result is \(\frac{1}{125}\).

Step 2: Apply the Logarithmic Formulation

The general form of a logarithmic equation is \(\log_{b}(a) = c\), which is equivalent to the exponential form \(b^c = a\). Based on the given equation \(5^{-3} = \frac{1}{125}\), we can rewrite this as a logarithmic equation by expressing it as \(\log_{5}\left(\frac{1}{125}\right) = -3\).

Step 3: Write the Final Logarithmic Equation

Based on the relationship identified from the previous steps, the logarithmic form of the equation \(5^{-3} = \frac{1}{125}\) is \(\log_{5}\left(\frac{1}{125}\right) = -3\).

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a constant base is raised to a variable exponent. These equations are vital in various fields such as science, engineering, and finance. Exponential equations take the form of \( b^c = a \), where:

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

For instance, in the equation \( 5^{-3} = \frac{1}{125} \), 5 is the base, -3 is the exponent, and \( \frac{1}{125} \) is the result. The challenge often lies in understanding and manipulating these components to solve for unknown exponents or bases. It's essential to identify these parts correctly to convert the equation accurately into another form, such as a logarithm.

Logarithms

Logarithms are the inverse operation of exponentiation. They help us determine the exponent needed to produce a given number from a specific base. In mathematical terms, if we know \( b^c = a \), then the logarithm form will be \( \log_{b}(a) = c \). Logarithms answer the question, "To what exponent do we need to raise the base \( b \) to get the number \( a \)?" Logarithms are widely used due to their ability to simplify complex calculations, especially when solving equations involving exponential growth or decay. Whether dealing with population growth, radioactive decay, or financial interest calculations, understanding logarithms is essential.

Conversion from Exponential to Logarithmic Form

Converting between exponential and logarithmic forms is a fundamental skill in algebra. This conversion shifts an equation from being multiplicative to an additive perspective, often simplifying the solving process.Let's look at the conversion process using our example: \( 5^{-3} = \frac{1}{125} \). Here’s how we do it:

• Identify the base \( b \), which is 5.
• Recognize the exponent \( c \), which is -3.
• Determine the result \( a \), which is \( \frac{1}{125} \).

To convert this equation to logarithmic form, rearrange it to \( \log_{5}(\frac{1}{125}) = -3 \). This notation tells us what power 5 must be raised to obtain \( \frac{1}{125} \), which is -3, linking the two mathematical forms seamlessly.Understanding this conversion enhances problem-solving strategies and allows deeper insight into the relationships between exponential growth and analytical solutions.