Problem 29 - College Algebra with Corequisite Support

Question

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. \(\log \left(\frac{1}{100}\right)=-2\)

Step-by-Step Solution

Answer

The exponential equation is \(10^{-2} = \frac{1}{100}\).

Step 1: Understand the Logarithmic Form

The logarithmic equation given is \( \log \left( \frac{1}{100} \right) = -2 \). This equation is in the form \( \log_b(a) = c \). Here, \( b \) is the base, \( a \) is the argument, and \( c \) is the logarithm result.

Step 2: Identify the Base of the Logarithm

In common logarithms, which use "log" without a base, the base is implicitly \( 10 \). Thus, \( b = 10 \) for this equation.

Step 3: Convert Logarithmic to Exponential Equation

Using the definition of a logarithm, \( \log_b(a) = c \) can be rewritten as the exponential form \( b^c = a \). Substituting our values, we have \( 10^{-2} = \frac{1}{100} \).

Step 4: Verify Exponential Form

Confirm the conversion by checking the exponential calculation: \( 10^{-2} = \frac{1}{10^2} = \frac{1}{100} \), which verifies the exponential equation as accurate.

Key Concepts

Exponential Equations

Exponential equations are essential tools in mathematics, often arising in situations where variables are found as exponents. Understanding them is vital, particularly when dealing with logarithms. In these equations, a number called the "base" is raised to a power, or exponent, creating a new value. For example, in the equation \( b^c = a \), "\( b \)" is the base, "\( c \)" is the exponent, and "\( a \)" is what results from this expression.

• Example: Consider the equation \( 10^{-2} = \frac{1}{100} \). Here, "10" is the base and "-2" is the exponent.
• Exponential equations can often be rewritten in a logarithmic form, providing a different perspective or a simpler way to solve the equation.

By understanding how to toggle between exponential and logarithmic forms, such as converting \( \log_{10}(\frac{1}{100}) = -2 \) to \( 10^{-2} = \frac{1}{100} \), you gain a more profound and versatile grasp of problem-solving in algebra.

Base of Logarithm

The base of a logarithm is crucial in understanding and working with logarithmic equations. It represents the number which, raised to a power, results in the argument of the logarithm. Knowing the base allows us to convert between logarithmic and exponential forms effortlessly.

• For common logarithms, the base is always 10, which is often implicitly understood when no base is indicated. For example, \( \log(\frac{1}{100}) \) implies a base of 10, hence \( \log_{10}(\frac{1}{100}) \).
• The base dictates the "scaling factor" of growth in an exponential context - a base larger than 1 indicates growth, and a base between 0 and 1 indicates decay.

Understanding the base is integral to accurately converting logarithmic forms into exponential equations. In mathematical practice, such as in our original exercise, recognizing the base simplifies the path to finding solutions and verifying results.

Common Logarithms

Common logarithms are those that use 10 as their base. They are widely utilized in real-world applications, such as calculating the intensity of sound in decibels or measuring pH in chemistry.

• When a logarithm is written as "log" without specifying the base, it is assumed to be base 10. This leads to more straightforward computations and problem-solving.
• For educational exercises, common logarithms offer an approachable way to learn about logarithmic functions due to their simplicity in practice.

Using common logarithms, we can quickly write an equation like \( \log(\frac{1}{100}) = -2 \) and convert it into exponential form, \( 10^{-2} = \frac{1}{100} \), with ease. Mastering them provides a foundation for tackling more complex logarithmic functions.