Problem 149 - Precalculus

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

\(x = \frac{1}{8}\)

Step 1: Understanding The Problem

We are tasked with solving the equation \(\log_{2}(x) = -3\) for \(x\). The logarithmic notation tells us that \(x\) is the result achieved when \(2\) is raised to the power of \(-3\).

Step 2: Converting Logarithmic to Exponential Form

Recall that \(\log_b(a) = c\) implies \(b^c = a\). For our problem, \(\log_{2}(x) = -3\) implies \(2^{-3} = x\). Thus, the equation in exponential form is \(x = 2^{-3}\).

Step 3: Calculating the Exponential Form

Now calculate \(x = 2^{-3}\). Use the properties of exponents: \(2^{-3} = \frac{1}{2^3}\). Calculate \(2^3 = 2 \times 2 \times 2 = 8\). Thus, \(2^{-3} = \frac{1}{8}\), so \(x = \frac{1}{8}\).

Key Concepts

Logarithmic Equation

A logarithmic equation involves a logarithm containing a variable. These types of equations are commonly solved by using properties of logarithms, which allow you to express the relationship between numbers in terms of exponents. Logarithms are an inverse operation to exponentiation, just like division is to multiplication. For example, in the equation \( \log_{2}(x) = -3 \), the number 2 is known as the base of the logarithm, and \(-3\) is the logarithm value. Here, the variable \(x\) is what you need to find. Understanding logarithms as exponents helps in solving such equations, where a given number is expressed in terms of its base raised to another number or exponent.

Converting Logarithms to Exponents

To solve a logarithmic equation, you often convert it into an exponential form. This is because dealing with numbers as exponents can sometimes simplify the complexity of the logarithmic equation. The general rule is: if \( \log_b(a) = c \), this is equivalent to saying \( b^c = a \). By following this rule, you can translate any logarithmic equation into an exponential form. Let's look at the equation from our example: \( \log_{2}(x) = -3 \). In exponential form, this becomes \( 2^{-3} = x \). What this means is, we are asking, 'what power must 2 be raised to in order to get x?'. By converting and rewriting the equation as \( 2^{-3} \), you can directly see that \( x \) must be \( \frac{1}{8} \). Converting logarithmic equations to exponential form is a powerful technique in algebra for isolating variables and solving for unknowns.

Solving for x in Logarithmic Equations

Once a logarithmic equation is converted to its exponential form, solving for the variable becomes straightforward. In the case of our example, after converting \( \log_{2}(x) = -3 \) to \( 2^{-3} = x \), the next step is to calculate the expression. To solve for \( x \) in this equation, you compute \( 2^{-3} \). Remember, \( 2^{-3} \) means 1 over \( 2^3 \). Calculating \( 2^3 \) gives us 8, because \( 2 \times 2 \times 2 = 8 \). Therefore, \( 2^{-3} \) becomes \( \frac{1}{8} \). Consequently, the solution for \( x \) is \( \frac{1}{8} \). Solving equations like these requires a solid understanding of exponents and logarithmic conversions; once mastered, these skills will assist in tackling more complex mathematical problems.