Problem 59 - Algebra 2

Question

Write each equation in exponential form. $$ \log _{2} \frac{1}{2}=-1 $$

Step-by-Step Solution

Answer

The exponential form of the given logarithmic equation \(\log_{2} \frac{1}{2} = -1\) is \(2^{-1} = \frac{1}{2}\).

Step 1: Identify the base, exponent, and result

From the logarithmic equation \(\log_{2} \frac{1}{2} = -1\), identify the base, the log result, and the log equals to number. Here, 2 is the base, -1 is the exponent, and \(\frac{1}{2}\) is the result.

Step 2: Write the equation in exponential form.

Using the correspondence between logarithmic and exponential forms, write the equation in its equivalent exponential form. The conversion follows the structure \(b^{n} = m\). Here \(b = 2\), \(n = -1\), and \(m = \frac{1}{2}\). Therefore the exponential equivalent of the logarithmic equation is \(2^{-1} = \frac{1}{2}\).

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical sentences that involve logarithms, expressions to express how many times a specific number, called the base, must be multiplied by itself to reach another number. In a simple logarithmic equation, such as \(\log_{2}\frac{1}{2} = -1\), each part has a specific role. The expression means that the base number (2) raised to the power of -1 equals \(\frac{1}{2}\). This particular form, \(\log_b (m) = n\), is telling us the exponent \(n\) to which the base \(b\) is raised to result in the number \(m\).

Logarithms have widespread applications in various fields, such as science, engineering, and finance, making it important to understand how to manipulate these equations efficiently. Understanding how to convert between logarithmic and exponential forms is a key skill in solving problems involving growth and decay, sound intensity, pH in chemistry, and more.

Base and Exponent

In mathematics, the base and the exponent are the fundamental components of exponential expressions. The base is the number that is repeatedly multiplied, and the exponent indicates how many times the base is used in the multiplication.

- In the expression \(b^n\), \(b\) is the base.- \(n\) is the exponent.

For example, in our initial equation \(\log_{2}\frac{1}{2} = -1\), the number 2 is the base and -1 is the exponent. Writing in exponential form, this becomes \(2^{-1} = \frac{1}{2}\). The negative exponent means that instead of multiplying, you take the reciprocal of the base. Therefore, \(2^{-1} = \frac{1}{2}\), which demonstrates that 2 needs to be inverted to achieve 0.5, or one half, in a single step.

Exponential Conversion

Converting from a logarithmic form to an exponential form involves identifying the base, the result, and the exponent, just as we demonstrated with the equation \(\log_{2}\frac{1}{2} = -1\). This conversion process is crucial for solving logarithmic equations and understanding the relationship between the logarithmic and exponential expressions.

Here's how you perform an exponential conversion:

• Identify the base \(b\), which is the number after 'log'.
• Identify the result \(m\), which is the number inside the log.
• Identify the exponent \(n\), which is on the other side of the equation.

Using this data, you write in the format \(b^{n} = m\). So for \(\log_{2}\frac{1}{2} = -1\), the exponential form is \(2^{-1} = \frac{1}{2}\).

Understanding this conversion helps in making sense of various real-world dynamics like computing compound interest, understanding radioactive decay, and many physical phenomena present in our universe.