Problem 26 - Algebra 2

Question

Write each equation in exponential form. \(\log _{100} \frac{1}{10}=-\frac{1}{2}\)

Step-by-Step Solution

Answer

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

Step 1: Understanding the Logarithmic Equation

The logarithmic equation given is \( \log_{100} \frac{1}{10} = -\frac{1}{2} \). Here, the base of the logarithm is 100, the argument is \( \frac{1}{10} \), and the result of the logarithm is \( -\frac{1}{2} \). We need to rewrite this in exponential form.

Step 2: Applying the Definition of Logarithms

Recall that the definition of logarithm \( \log_b a = c \) is equivalent to the exponential form \( b^c = a \). In our equation, \( b = 100 \), \( a = \frac{1}{10} \), and \( c = -\frac{1}{2} \).

Step 3: Writing in Exponential Form

Using the equivalence from the definition, rewrite the logarithmic form \( \log_{100} \frac{1}{10} = -\frac{1}{2} \) to an exponential equation: \( 100^{-\frac{1}{2}} = \frac{1}{10} \). This shows that raising the base 100 to the power of -\( \frac{1}{2} \) gives \( \frac{1}{10} \).

Key Concepts

Logarithmic Equations

Logarithmic equations are a key component in mathematics that help us deal with exponential relationships. A logarithmic equation is essentially an equation that involves the logarithm of a variable or number. In simpler terms, it helps us find out what power a number must be raised to, in order to obtain another number. For example, in the equation \( \log_{100} \frac{1}{10} = -\frac{1}{2} \), we are determining the power to which the base number 100 must be raised to produce the fraction \( \frac{1}{10} \).
To effectively work with these equations, it's crucial to first identify the parts:

• The logarithm (\( \log \)) indicates the operation.
• The base (just after \( \log \)) is the number we are raising to a power.
• The argument (following the base) is the result we are interested in matching.
• The solution (on the other side of the equation) is the exponent we need.

These equations are solved by employing transformation techniques that convert them into exponential form, allowing us to better understand and solve for variables involved.

Base of Logarithms

Understanding the base of a logarithm is crucial when working with logarithmic and exponential functions. The base in a logarithmic equation, such as in \( \log_{100} \frac{1}{10} = -\frac{1}{2} \), is the "100". This base is the number that we repeatedly multiply in an exponential manner.
For instance, in this equation, the base 100 is the number we seek to raise to a certain power to equal \( \frac{1}{10} \).

• A key property: the logarithm base must be greater than zero and not equal to one.
• When you know the logarithmic values, it is often easier to find complex exponential solutions through conversion.

Different bases can radically change the outcome of the equation if we were to raise them to a particular exponent. Therefore, a clear understanding of the base allows you to accurately convert between logarithmic and exponential forms.

Converting Logarithmic to Exponential

The process of converting a logarithmic equation into its corresponding exponential form is essential for solving logarithmic problems. This transformation follows directly from the definition of logarithms. The standard form of a logarithmic equation \( \log_b a = c \) can always be written in exponential form \( b^c = a \).
In the exercise \( \log_{100} \frac{1}{10} = -\frac{1}{2} \), we transformed it into exponential form as \( 100^{-\frac{1}{2}} = \frac{1}{10} \).

• One reason for this conversion is that many mathematical operations are simplified using exponentials.
• Exponential form also offers intuitive insights into the relationship between the involved quantities.

Thus, converting logarithms to exponentials is both a practical skill in mathematics and a fundamental concept that reinforces the interconnected nature of these two key mathematical operations.