Problem 44 - A Graphical Approach to College Algebra

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log x+\log x^{2}=3$$

Step-by-Step Solution

Answer

The solution is \( x = 10 \).

Step 1: Understand the Problem

The equation we need to solve is \( \log x + \log x^2 = 3 \). We have to find the value of \( x \) that satisfies this equation.

Step 2: Apply Logarithmic Properties

Use the logarithmic property \( \log a + \log b = \log(ab) \) to combine the logs. The equation becomes \( \log(x \cdot x^2) = \log(x^3) = 3 \).

Step 3: Convert Logarithmic to Exponential Form

By converting the logarithmic equation \( \log(x^3) = 3 \) to its exponential form, we get \( x^3 = 10^3 \).

Step 4: Solve for x

Since \( x^3 = 1000 \), take the cube root of both sides to solve for \( x \). This gives us \( x = \sqrt[3]{1000} \).

Step 5: Calculate the Solution

The cube root of 1000 is 10, so we find \( x = 10 \).

Step 6: Verify the Solution

Check the solution by substituting \( x = 10 \) back into the original equation: \( \log 10 + \log 10^2 = 1 + 2 = 3 \), which matches the right-hand side of the original equation.

Key Concepts

Logarithmic Properties

Logarithmic properties are essential tools for rewriting and simplifying equations involving logarithms. One crucial property used in the problem is the product rule of logarithms. This rule states that the logarithm of a product is equal to the sum of the logarithms of the factors. In formula terms, this can be expressed as: \( \log(a) + \log(b) = \log(a \cdot b) \).

• In this exercise, we combined \( \log(x) \) and \( \log(x^2) \) using the product rule, resulting in \( \log(x \cdot x^2) = \log(x^3) \).
• This simplification is useful because it reduces the equation to a single logarithmic form that is easier to manipulate.

Understanding and applying logarithmic properties not only aids in simplifying complex logarithmic expressions but also sets up the equation for further algebraic manipulation.

Exponential Form

Converting logarithmic equations into exponential form is a critical step in solving them. When you have a logarithmic equation such as \( \log_b (a) = c \), you can switch to an exponential form: \( a = b^c \). This conversion is based on the definition of a logarithm, which tells us that a logarithm answers the question: "To what power must the base \( b \) be raised, to produce \( a \)?"

• In our problem, we reached the equation \( \log(x^3) = 3 \). To convert this into an exponential form, we raised the base, which is 10 (since logarithms without a base written are typically base 10), to the power of 3, resulting in \( x^3 = 10^3 \).
• This transformation makes it possible to proceed with solving for \( x \) using basic algebraic techniques.

By mastering the technique of switching between logarithms and exponentials, students can tackle a wider range of equations effectively.

Solving for x

Once the logarithmic expression is converted into exponential form, the actual value of \( x \) can be determined through algebraic manipulation. In our example, after establishing that \( x^3 = 1000 \), the next step was to solve for \( x \) by finding the cube root of both sides.

• The equation \( x^3 = 1000 \) leads directly to \( x = \sqrt[3]{1000} \), which simplifies further to \( x = 10 \), since \( 10^3 = 1000 \).
• Finding the cube root can often involve straightforward arithmetic, though it's important to be comfortable with using a calculator for more complex figures.
• Verification is also crucial. By substituting \( x = 10 \) back into the original logarithmic equation, we confirmed that the equation holds true, ensuring the solution is correct.

Understanding these steps solidifies the student’s grasp on solving logarithmic equations, providing a pathway to confidently tackle similar problems.