Problem 46 - Precalculus

Question

Solve the logarithmic equations exactly. $$\log x^{2}+\log x=3$$

Step-by-Step Solution

Answer

The solution is \(x = 10\).

Step 1: Use Logarithm Rules

The problem involves two logarithmic expressions added together: \(\log x^2 + \log x\). According to logarithmic rules, specifically the product rule, this can be written as a single logarithm: \(\log(x^2 \cdot x) = \log x^3\). So the equation becomes \(\log x^3 = 3\).

Step 2: Convert Logarithmic to Exponential Form

To solve \(\log x^3 = 3\), convert the logarithm to its exponential form. Since \(\log\) has a base of 10 (common logarithm), we have \(x^3 = 10^3\).

Step 3: Solve the Exponential Equation

Now, solve for \(x\) in the equation \(x^3 = 10^3\). Calculating \(10^3\) gives 1000, so \(x^3 = 1000\). Take the cube root of both sides: \(x = \sqrt[3]{1000}\). Calculating the cube root gives \(x = 10\).

Step 4: Verify the Solution

Substitute \(x = 10\) back into the original equation to check the solution. Calculate \(\log(10^2) + \log(10) = \log(100) + \log(10)\). Using the product rule, this is \(\log(1000)\), which is indeed 3, confirming that \(x = 10\) is the correct solution.

Key Concepts

Logarithm Rules

Understanding the rules that govern logarithms is key to solving many mathematical problems. In this exercise, we encounter two crucial rules: the product rule and the power rule.

The product rule states that the sum of two logarithms with the same base is equivalent to the logarithm of the product of their arguments. For example,

• \( \log_a (m) + \log_a (n) = \log_a (m n) \)

This rule simplifies complex logarithmic equations by combining separate logarithmic terms into one.

Another important rule is the power rule, which allows us to simplify expressions where a logarithm is taken of a power. It states that

• \( \log_a (m^n) = n \log_a (m) \)

In our exercise, these rules help convert the expression \( \log x^2 + \log x \) into \( \log x^3 \), setting the stage for easier manipulation and solution.

Exponential Form

Knowing how to convert logarithmic equations into exponential form can greatly simplify problem-solving. In our exercise, starting with an equation \( \log x^3 = 3 \), we utilize the definition of logarithms.

• In terms of base 10, since the equation is in the form \( \log_{10} (x^3) = 3 \),
• We convert this to its exponential counterpart: \( x^3 = 10^3 \).

This conversion reveals the relationship between the terms:

• The base (10) raised to the logarithm's resulting value equals the original argument of the logarithm (\( x^3 \)).

Once expressed in exponential form, the equation becomes straightforward to solve, bringing us closer to finding the desired variable.

Product Rule

The product rule for logarithms, which simplifies the sum of two or more logarithmic terms, is particularly powerful. As seen in the exercise, we start with \( \log x^2 + \log x \). Applying the product rule:

• Combine the two logarithmic expressions into one single logarithm:
• \( \log x^2 + \log x = \log(x^2 \cdot x) = \log x^3 \).

The essence of the product rule lies in transforming what might seem to be a complex expression into a more manageable form with a single logarithmic term.

The simplification achieved by the product rule not only makes it easier to work with the equation, but also clearly reveals possible solutions, such as in the subsequent step of exponentiating to solve for \( x \). This rule often serves as the first step in breaking down intricate logarithmic expressions for smoother problem-solving.