Question

\(\begin{aligned} \text { Find } x \text {, if } \log x^{3}-\log 3 x &=2 \log 2+\log 3: \\ & \text { (b) } 6 \end{aligned}\) (b) 6 (a) 3 (d) none of these (c) 9

Short answer

Based on the step by step solution provided above, the correct answer is: (d) None of these

Step-by-step solution

Combine the logarithms with subtraction using the division property of logarithms

We are given that $$\log x^{3}-\log 3x = 2\log2+\log 3.$$ Using the division property of logarithms, which states that \(\log a - \log b = \log(a / b)\), we can combine the left side: $$\log \frac{x^3}{3x} = 2\log 2+\log 3.$$

Simplify the expression inside the logarithm

The expression inside the logarithm can be simplified: $$\log \frac{x^3}{3x} = \log \frac{x^3}{x(3)}=\log x^2.$$ Now our equation is: $$\log x^2 = 2\log 2+\log 3.$$

Combine the logarithms with addition using the multiplication property of logarithms

Using the multiplication property of logarithms, which states that \(\log a + \log b = \log(ab)\), we can combine the right side: $$\log x^2 = \log (2^2 \cdot 3).$$ Simplify the right side: $$\log x^2 = \log (12).$$

Solve for x

Since the logs on both sides are equal, we can drop the logarithms and solve for x: $$x^2 = 12.$$ Taking the square root of both sides, $$x = \pm\sqrt{12}.$$ However, since we're looking for a positive value for x, we choose the positive square root. So, \(x = \sqrt{12}.\) Since \(\sqrt{12}\) is not among the given options, the correct answer is (d) none of these.

Key concepts

Division Property of Logarithms

The division property of logarithms is a handy tool when we're dealing with logarithmic expressions that include subtraction. This property simplifies the process by consolidating two separate logarithms into one. To apply this property, remember the rule:

• If you have \(\log a - \log b\), it becomes \(\log \frac{a}{b}\).

This principle is invaluable for simplifying expressions and solving equations.
For instance, in the given exercise, we're evaluating \(\log x^3 - \log 3x\). Applying the division property, you transform this into one log: \(\log \frac{x^3}{3x}\). This simplifies the expression significantly and makes it more manageable.
When working with logarithmic equations, utilizing the division property efficiently can pave the way for easier solutions by decreasing the number of terms you need to handle. It helps to bring everything down to a single logarithm, putting the equation in a form that's easier to solve.

Multiplication Property of Logarithms

The multiplication property of logarithms allows you to combine logarithmic expressions that involve addition. This property simplifies expressions, much like its division counterpart. The golden rule here is:

• For expressions like \(\log a + \log b\), you can combine them into one: \(\log(ab)\).

This property is particularly useful in equations, where simplifying the terms can bring clarity and a path to solve the problem more easily.
In our step-by-step exercise solution, we used it to transform \(2\log 2 + \log 3\) on the right side of the equation. We apply the multiplication property to get \(\log ((2^2) \cdot 3)\), which simplifies further to \(\log 12\).
By using the multiplication property, solving the equation becomes a more straightforward task, as you're reducing the bulk of the expression, making it easier to equate different parts of the equation.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions is an essential skill that involves the application of logarithmic properties and rules to make complex expressions more manageable. Here's how you can approach it:

• First, identify which properties of logarithms apply to the given expression. These could be addition, subtraction, or another log property.
• Apply the division or multiplication property as necessary to consolidate the terms into a simpler form.
• Ensure all like terms are combined; this often leads to fewer terms and a cleaner expression.

In our exercise, simplifying was key. We used simplification alongside the division property to go from \( \log \frac{x^3}{3x} \) to \( \log x^2 \).
On the right side, applying simplification after using the multiplication property turned \(2\log 2 + \log 3\) into \( \log 12 \).
Ultimately, the simplification of log expressions can make difficult equations accessible, allowing for easier solution derivation by focusing only on the necessary parts.