Problem 22 - Precalculus with Calculus Previews

Question

Solve the given logarithmic equation. $$ \ln (10+x)=\ln (3+4 x) $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = \frac{7}{3} \).

Step 1: Set the equations equal to each other

Since the natural logarithms are already equal, we can eliminate the logarithms by setting the expressions inside the logarithms equal to each other. This gives us: \[ 10 + x = 3 + 4x \]

Step 2: Rearrange the equation

Move the term involving \(x\) from the right side to the left by subtracting \(x\) from both sides:\[ 10 + x - x = 3 + 4x - x \]which simplifies to:\[ 10 = 3 + 3x \]

Step 3: Isolate the variable term

Subtract 3 from both sides to isolate the term with \(x\):\[ 10 - 3 = 3x \]which simplifies to:\[ 7 = 3x \]

Step 4: Solve for x

Divide both sides by 3 to solve for \(x\):\[ x = \frac{7}{3} \]

Key Concepts

Understanding Natural Logarithms

Natural logarithms are a specific type of logarithm that is written with the symbol \( \ln \), which stands for the base \( e \) logarithm. This base \( e \) is an irrational constant approximately equal to 2.71828.
It is a crucial mathematical constant in various fields such as calculus, complex numbers, and physics. Natural logarithms follow the same basic rules as other logarithms, but are specifically focused on the base \( e \).
When working with equations like \( \ln(10 + x) = \ln(3 + 4x) \), knowing that their bases are the same allows us to concentrate on the expressions inside the logarithms.
By leveraging the properties of logarithms, we can directly set these expressions equal to each other: \( 10 + x = 3 + 4x \). This simplification is possible because if \( \ln(a) = \ln(b) \), then \( a = b \). Understanding this property is critical in solving logarithmic equations efficiently.

Isolating Variables in Equations

Once we remove the logarithms from our equation, we encounter a linear equation \( 10 + x = 3 + 4x \). The next step is to isolate the variable. Isolating variables makes it simpler to solve the equation because it involves getting the variable on one side of the equation.
One efficient strategy for isolating variables is to collect all the terms containing the variable on one side and constant terms on the other.
In our equation, it starts by moving terms with the variable \( x \) onto one side, while constants move to the opposite side, like this:
\( 10 + x - x = 3 + 4x - x \). This simplification leads to \( 10 = 3 + 3x \).
This simple rearrangement forms the foundation of isolating variables, allowing us to only focus on the term we need to find.

Solving Linear Equations

Solving linear equations usually involves a straightforward methodology once variables are isolated. In our example, we ended up with \( 10 = 3 + 3x \).
The subsequent step is to further simplify the equation by isolating \( x \) completely. We accomplish this by subtracting the constant term from both sides, leading us to \( 10 - 3 = 3x \) which simplifies to \( 7 = 3x \).
Finally, we divide each side of the equation by the coefficient of \( x \) (which is 3 in this case), arriving at \( x = \frac{7}{3} \). Using such step-by-step processes builds essential skills in solving linear equations efficiently.

• This involves operations such as addition, subtraction, multiplication, and division.
• It also reinforces careful manipulation of equations to maintain their balance.

Understanding these steps allows us to solve any linear equation systematically and confidently.