Problem 66 - Algebra 2

Question

Solve each equation. $$ \ln x^{2}+\ln x=6 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \sqrt[3]{e^6}\)

Step 1: Use properties of logarithms to simplify

Using property of logarithms, the sum of two log terms of the same base is equivalent to the log of the product of the terms. Hence, the equation will simplify as follows: \[ \ln (x^2 \cdot x) = 6 \]

Step 2: Simplify the log term

Now, simplify the product inside the logarithm: \[ \ln (x^3) = 6 \]

Step 3: Convert to exponential form

Since the equation is in the form 'ln base = exponent', convert it to exponential form to solve for x. Using definition of natural logarithm, the equation translates to: \[ e^6 = x^3 \]

Step 4: Solve for x

Taking cubed root on both sides gives, \[ x = \sqrt[3]{e^6} \] This is valid as long as the value is positive since logarithms are undefined for non-positive numbers.

Key Concepts

Properties of Logarithms

In the study of logarithmic equations, an understanding of the properties of logarithms is essential. These properties make it easier to simplify complex logarithmic expressions and solve equations efficiently. When dealing with the logarithm of a product, you can apply the product property of logarithms. This property states that the logarithm of a product is equal to the sum of the logarithms of each factor. For instance, if you have \[ \ln a + \ln b \] you can combine these into one logarithm, \[ \ln(a \cdot b). \] In the given exercise, this property was used to combine \[ \ln x^2 + \ln x \] into \[ \ln(x^2 \cdot x). \] By applying these rules, calculations become more straightforward, maintaining a neat and clear equation to solve further steps.

Exponential Form

The exponential form of an equation is critical in converting logarithmic equations into a solvable algebraic form. The concept stems from the definition of a logarithm. Essentially, if \( \ln(b) = c \), then the equivalent exponential form is \( e^c = b \). In our exercise, we reached a point where: \[ \ln(x^3) = 6. \] Converting this into exponential form means raising the base of the natural logarithm, which is \( e \), to the power on the right-hand side of the equation. Thus, the equation becomes: \[ e^6 = x^3. \] This conversion is vital as it transforms a logarithmic equation into a familiar algebraic expression which many find easier to solve.

Solving Algebraic Equations

Once you've transformed a logarithmic equation to its exponential form, solving it relies on algebraic principles to isolate and find the unknown variable. In our example, after converting to exponential form, we had: \[ e^6 = x^3. \] To solve for \( x \), we need to eliminate the cube by taking the cubed root on both sides of the equation. This gives: \[ x = \sqrt[3]{e^6}. \] It is important to remember that logarithms are defined only for positive numbers, hence ensuring \( x \) must be positive. Therefore, whenever encountering such an equation, focus on steps that isolate the variable and ensure any solution is permissible within the context of the original logarithmic expression.