Problem 14 - Algebra 2

Question

Solve each equation. Check your answers. $$ \ln 3 x=6 $$

Step-by-Step Solution

Answer

The solution to the equation \( \ln 3x = 6 \) is \( x = e^6 / 3 \).

Step 1: Isolate the logarithm

The given equation is \( \ln 3x = 6 \). Here, the natural logarithm, ln, is already isolated on one side of the equation.

Step 2: Transform into exponential form

We can transform the logarithmic equation into an equivalent exponential equation using the definition of the logarithm. This gives \( e^6 = 3x \). Here, \( e \) is the base of the natural logarithm.

Step 3: Solve for x

The final step is to solve the exponential equation for x. By dividing both sides of the equation by 3, we find the solution: \( x = e^6 / 3 \). We can use a calculator to compute the exact value of x, if required, however expressing the answer in terms of \( e \) is perfectly acceptable.

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical expressions that involve a logarithm with a variable. These equations are powerful tools that allow us to solve for unknown values in expressions where the variable is part of a logarithm. In general, a logarithmic equation has the form \( \log_b(x) = y \). Here, \( b \) is the base of the logarithm, \( x \) is the argument (the variable part), and \( y \) is the value of the logarithm.

To solve logarithmic equations, we often transform them into exponential equations, since logarithms and exponents are inverse operations. This transformation helps to simplify the process of finding the value of the variable.

• Identify and isolate the logarithm on one side of the equation.
• Convert the logarithmic equation into an exponential form using the basic logarithmic identity: \( \log_b(x) = y \) becomes \( x = b^y \).
• Solve the resulting exponential equation for the unknown variable.

This method makes it easier to solve these equations and find the values that satisfy the original logarithmic expression.

Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a special type of logarithm where the base is the mathematical constant \( e \,\approx 2.71828 \). The natural logarithm is widely used in mathematics and science because of its natural occurrence in various phenomena, especially those involving growth and decay.

The natural logarithm \( \ln(x) \) can be expressed in exponential form as \( e^y = x \), where \( y \) is the logarithmic value of \( x \). For instance, if \( \ln(3x) = 6 \,\) we can solve for \( 3x \) by transforming it into an exponential equation: \( 3x = e^6 \).

• The natural logarithm is particularly useful for continuous compounding and growth processes.
• It simplifies differentiation and integration of exponential functions.
• \( \ln(x) \) is undefined for \( x \leq 0 \).

In solving equations involving natural logarithms, being comfortable with transforming between logarithmic and exponential forms is essential for simplifying complex problems.

Exponential Form

Exponential form converts a logarithmic equation into a form that involves exponents, which can make solving the equation more intuitive. For the equation \( \ln(3x) = 6 \,\) converting it to exponential form gives \( e^6 = 3x \). This change is based on understanding that the logarithm and exponent are inverse operations.

This transformation helps to solve equations that are otherwise difficult to tackle directly in their logarithmic form.

• Use the property: \( \log_b(x) = y \rightarrow x = b^y \).
• For natural logarithms, specifically \( \ln(x) = y \rightarrow x = e^y \).
• This conversion sheds light on the relationship between the components of the original equation.

By recognizing and applying the exponential form, you can effectively manipulate and solve mathematical statements that include logarithms.

Isolating the Variable

Isolating the variable is a fundamental technique in solving equations. It simplifies complex expressions by rearranging the equation to make the variable of interest the subject of the equation. In the context of logarithmic or exponential equations, the goal is to express the variable in terms of other known quantities.

In our example, once the given logarithmic equation \( \ln(3x) = 6 \) is transformed to its exponential form \( e^6 = 3x \,\) the next step involves isolating \( x \.\) This is done by dividing both sides of the equation by 3, resulting in \( x = \frac{e^6}{3} \.\)

• Begin by simplifying the equation as much as possible.
• Perform operations (addition, subtraction, multiplication, division) that keep the equation balanced while isolating the variable.
• Check your work by substituting the value of the variable back into the original equation to verify the solution.

Mastering the art of isolating variables will greatly enhance your problem-solving skills and is applicable across various mathematical disciplines.