Problem 8 - Algebra 2

Question

Write each equation in exponential form. \(\log _{2} \frac{1}{8}\)

Step-by-Step Solution

Answer

The exponential form is \(2^{-3} = \frac{1}{8}\).

Step 1: Understand the Given Logarithmic Expression

The given expression is \(\log_{2} \frac{1}{8}\). This means that we are looking for an exponent \(x\) such that when 2 is raised to this power, the result is \(\frac{1}{8}\). In exponential form, this is written as \(2^x = \frac{1}{8}\).

Step 2: Write the Fraction as a Power of 2

Noticing that \(\frac{1}{8}\) can be rewritten as a power of 2, we observe that \(8 = 2^3\) and thus \(\frac{1}{8} = 2^{-3}\). This gives us \(2^x = 2^{-3}\).

Step 3: Solve for the Exponent

Since both sides of the expression \(2^x = 2^{-3}\) have the same base, we can set the exponents equal to each other. Thus, \(x = -3\).

Step 4: Write in Complete Exponential Form

From the previous steps, the exponential form of the expression \(\log_{2} \frac{1}{8}\) is \(2^{-3} = \frac{1}{8}\).

Key Concepts

Exponential Form

The exponential form of an equation is a way of expressing numbers using exponents. It helps to clarify the relationship between the base, the exponent, and the result of raising the base to that exponent. When you have a logarithmic statement, like \( \log_{2} \frac{1}{8} \), it describes a scenario where you're looking for the exponent that makes the base (in this case, 2) equal to a given number (\( \frac{1}{8} \)). Writing this in exponential form, the logarithmic equation \( \log_{2} \frac{1}{8} = x \) converts to \( 2^x = \frac{1}{8} \).

• This process involves recognizing the components of the logarithmic statement (base, power/result).
• Transforming it for clear understanding, especially when transitioning to solve or interpret logarithmic expressions.

By expressing the equation in exponential form, it becomes easier to visualize and solve the equation as it translates into a direct comparison of powers.

Base of Logarithm

Understanding the base of logarithms is crucial when working with logarithmic expressions. The base is the number that is repeatedly multiplied by itself to get another number. In the context of the logarithmic equation \( \log_{2} \frac{1}{8} \), the base is 2.It indicates that we seek the power to which 2 must be raised to produce the number \( \frac{1}{8} \). Analyzing this, we can convert it into the equivalent exponential form which is \( 2^x = \frac{1}{8} \).

• A base in logarithms determines the factor of multiplication.
• It's a critical factor in computing logarithms and transitioning to exponential expressions.

Choosing the correct base and understanding how it factors into the equation can simplify and demystify the process, facilitating the conversion and comprehension of logarithmic equations.

Power of a Number

The power of a number indicates how many times you multiply the number by itself. In equations like \( 2^x = \frac{1}{8} \), which is derived from \( \log_{2} \frac{1}{8} \), the power is denoted by \( x \).This power \( x \) is the unknown we are solving for. In examining the equation, recognizing that \( \frac{1}{8} \) can also be written as a power of 2 with a negative exponent is key. Specifically, since \(8\) is \(2^3\), then \( \frac{1}{8} \) becomes \(2^{-3}\).Hence, equating base-with-power representations leads us to \( 2^x = 2^{-3} \), where the exponents, \( x = -3 \), are directly compared.

• The process involves identifying equivalent powers.
• Understanding negative powers can truly unlock the solution, especially in reverse multiplication by fractions.

Knowledge of how powers, especially negative ones, work with the base number allows for precise and straightforward solutions to logarithmic problems.