Problem 77 - Intermediate Algebra

Question

Find each value of \(x .\) $$ \log _{x} 1,000=3 $$

Step-by-Step Solution

Answer

The value of \( x \) is 10.

Step 1: Understand the Logarithmic Equation

We start with the given equation \( \log_{x} 1000 = 3 \), which means that the base \( x \) raised to the power of 3 equals 1000. We need to express this relationship as an exponential equation to find \( x \).

Step 2: Convert to Exponential Form

The logarithmic equation \( \log_{x} 1000 = 3 \) can be rewritten in its exponential form as \( x^3 = 1000 \). This tells us that \( x \) raised to the power of 3 equals 1000.

Step 3: Solve the Exponential Equation

To find \( x \), we need to take the cube root of both sides of the equation \( x^3 = 1000 \). This gives us \( x = \sqrt[3]{1000} \).

Step 4: Calculate the Cube Root

Calculate the cube root of 1000. We know that \( 10^3 = 1000 \), so \( \sqrt[3]{1000} = 10 \). Therefore, \( x = 10 \).

Key Concepts

Exponential Form

In this exercise, we transform the given logarithmic equation into an exponential form to make it easier to solve. A logarithm expresses one number as a power of another. For example, if you have \[\log_{x} 1000 = 3\]it means that the base \( x \) raised to the power of 3 is 1000. Thus, converting this into an exponential form helps us express this relationship clearly as:

• \( x^3 = 1000 \)

Switching between logarithmic and exponential forms is a fundamental concept in algebra, useful to solve for unknown values like \( x \). Remember:

• The base (\( x \)) raised to the result (3) equals 1000 in this case.
• This method simplifies complex logarithmic equations into more manageable exponential equations.

Cube Root

In solving \( x^3 = 1000 \), finding the cube root is crucial. A cube root of a number \( a \) is a number \( b \) such that \( b^3 = a \). Therefore, cube roots mean reversing the process of cubing a number. When given:\[x^3 = 1000\]we solve for \( x \) by applying the cube root to both sides:

• \( x = \sqrt[3]{1000} \)

Calculating the cube root involves:

• Identifying the number that multiplies by itself twice to get back to the original number, which is 1000 here.
• Recognizing that \( 10^3 = 1000 \).
• So, \( \sqrt[3]{1000} = 10 \).

Taking cube roots is a vital skill, enabling you to determine original values before they were cubed. It's also an essential part of various mathematical and real-world applications.

Solving Equations

The process of solving equations involves finding the unknown value that satisfies the equation's condition. This exercise highlights a common procedure:

• Understanding the initial logarithmic form \( \log_{x} 1000 = 3 \), interpreted as an exponential equation: \( x^3 = 1000 \).
• The next step was moving to solve for \( x \) through cube roots.
• Finally, we assessed the solution to ensure correctness by substituting back: \( 10^3 = 1000 \), verifying that \( x = 10 \) is indeed valid.

When solving equations:

• Carefully convert forms to the simplest equations (e.g., log to exponential).
• Apply mathematical operations (e.g., taking cube roots) to isolate and solve for unknowns.
• Always verify the solution by substituting it back into the original equation.

Solving equations is more than just finding \( x \); it's also about understanding mathematical relationships and validating each step along the way. This method ensures accuracy and solidifies mathematical comprehension.