Problem 27 - College Algebra

Question

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. $$ \log _{2}(x)=-3 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{1}{8} \).

Step 1: Understand the Logarithmic Equation

The equation given is \( \log_{2}(x) = -3 \). This reads as 'log base 2 of \( x \) equals -3'. Our goal is to find the value of \( x \).

Step 2: Convert to Exponential Form

To convert the logarithmic equation \( \log_{2}(x) = -3 \) to its exponential form, use the definition of a logarithm: \( \log_{b}(a) = c \) is equivalent to \( b^{c} = a \). Thus, \( 2^{-3} = x \).

Step 3: Simplify the Exponential Expression

Calculate \( 2^{-3} \) by simplifying the expression. \( 2^{-3} \) means \( \frac{1}{2^{3}} \), which is \( \frac{1}{8} \).

Step 4: Conclusion

The value of \( x \) is \( \frac{1}{8} \). Thus, the solution to the equation \( \log_{2}(x) = -3 \) is \( x = \frac{1}{8} \).

Key Concepts

Logarithmic Equation

A logarithmic equation expresses a relationship involving a function of a number to an exponent, known as a log. You see this in forms like \( \log_{base}(value) = exponent \). Here, the logarithmic equation \( \log_{2}(x) = -3 \) represents the power to which base 2 must be raised to result in \( x \). The key to understanding logarithmic equations is recognizing the interconnection between logs and exponents. A log essentially asks, "To what power must the base be raised, to yield this number?" This is a fundamental concept that allows us to resolve unseen values in expressions. Understanding this equation starts you on the path to converting it into something more familiar and actionable.

Convert to Exponential

Converting a logarithmic equation to an exponential form is a straightforward yet powerful technique in algebra. If you understand the basic formula of logarithms: \( \log_{b}(a) = c \), you can rewrite it as: \( b^{c} = a \). For our equation \( \log_{2}(x) = -3 \), this transformation implies answering: "What power must 2 be raised to, to result in x?" By using the definition, we convert to the exponential form:

• Base \(b\) is 2
• Exponent \(c\) is -3
• Resultant \(a\) is \(x\)

Thus, \( 2^{-3} = x \). This conversion is essential because it provides a direct expression that can be calculated with ease, avoiding any hidden complexities of logs. Instead, we work with clarity on the exponential front.

Simplify Exponential Expression

Simplifying an exponential expression involves straightforward arithmetic but is crucial for clarity and accuracy. Once in exponential form \( 2^{-3} = x \), evaluate \( 2^{-3} \). The negative exponent rule tells us: \( a^{-b} = \frac{1}{a^{b}} \). So, \( 2^{-3} \) can be simplified to \( \frac{1}{2^3} \). This leads to:

• Evaluate \( 2^3 \): means multiplying 2 by itself three times (\( 2 \times 2 \times 2 = 8 \))
• Therefore, \( \frac{1}{2^3} = \frac{1}{8} \)

Hence, the solution is \( x = \frac{1}{8} \). The simplicity of exponential form not only helps solve problems but also solidifies the conceptual understanding of logs and exponents, critical in further math applications.