Problem 62 - Algebra 2

Question

Solve each equation. $$ \ln 2 x=3 $$

Step-by-Step Solution

Answer

The solution to the equation \(\ln 2x = 3\) is \(x \approx 10.04\)

Step 1: Convert the Log Equation to an Exponential Equation

The logarithmic equation \(\ln 2x = 3\) can be converted into the exponential form as \(2x = e^3\). 'e' is the base of the natural logarithm, and it is approximately equal to 2.71828.

Step 2: Solve for x

Now we can simplify this equation by isolating x. We get \(x = e^3 / 2\)

Step 3: Calculate the Numerical Value

After simplifying the equation, we can find the numerical value for x by substituting the value of the base e (2.71) in the equation. The numerical value will be \(x \approx 10.04\)

Key Concepts

Exponential Equations

Exponential equations are equations where the variable is located in the exponent. Solving them often involves using logarithms to 'bring down' the exponent so that the variable can be isolated. For example, the equation \(2x = e^3\) is an exponential equation stemming from \(\ln 2x = 3\). Here, \(e\) represents a mathematical constant approximately equal to 2.71828. To solve \(2x = e^3\), it's necessary to understand that you can equate powers of \(e\) and then perform basic algebraic steps, such as dividing by 2 in this case, to isolate the variable \(x\). Understanding this allows us to handle equations involving exponential growth and decay, which are common in real-world applications such as interest calculations and population growth predictions.

Natural Logarithms

Natural logarithms are logarithms with a base of \(e\). The symbol usually used to denote a natural logarithm is \(\ln\). The natural logarithm of a number is the exponent that \(e\) must be raised to in order to equal that number. For instance, in the equation \(\ln 2x = 3\), we are interested in finding the \(2x\) value that, when the base \(e\) is raised to the power of 3, will result in it.

• \(\ln\) makes it easier to work with exponential equations, especially when the exponential equation involves the constant \(e\).
• They are commonly used in scientific and financial contexts to simplify calculations that involve growth or decay processes.

The consistency and properties of natural logarithms make them an essential tool in solving equations that feature exponential growth or change.

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of an unknown variable. In cases like \(\ln 2x = 3\), this means rewriting the logarithmic equation as an exponential equation, performing algebraic manipulations, and finally calculating the value of the variable. Here’s a summary of the process:

• Convert the logarithmic equation to an exponential form: For \(\ln 2x = 3\), you'd convert it to \(2x = e^3\).
• Isolate the variable: Rearrange the equation to solve for \(x\), resulting in \(x = \frac{e^3}{2}\).
• Compute the numerical value: Substitute \(e\) with approximately 2.71 if a numerical approximation is needed, such as calculating \(x \approx 10.04\).

Through practice and familiarity with these methods, solving equations becomes a structured path to finding solutions, whether they involve simple operations or more complex functions.