Problem 35 - A Graphical Approach to College Algebra

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (4 x)=1.5$$

Step-by-Step Solution

Answer

The solution for \( x \) is \( \frac{e^{1.5}}{4} \).

Step 1: Understand the problem

We need to solve the logarithmic equation \( \ln(4x) = 1.5 \). The solution will involve finding the value of \( x \) such that when \( 4x \) is substituted back into the natural logarithm \( \ln \), the result equals 1.5.

Step 2: Isolate the logarithmic expression

The given equation is already in the form we need, \( \ln(4x) = 1.5 \), so we can proceed to the next step which is to eliminate the logarithm to solve for \( x \).

Step 3: Exponentiate both sides

To remove the natural logarithm, we exponentiate both sides of the equation using base \( e \). This yields: \[ 4x = e^{1.5} \]This step uses the property that if \( \ln(y) = z \), then \( y = e^z \).

Step 4: Solve for x

Now, we solve for \( x \) by dividing both sides of the equation by 4:\[ x = \frac{e^{1.5}}{4} \]

Step 5: Evaluate using a calculator

Use a calculator to find the numerical value of \( e^{1.5} \). This gives a value of approximately 4.4817. Therefore, \[ x = \frac{4.4817}{4} \approx 1.1204. \] However, since the problem asks for exact form, we state the solution in exact terms as \( x = \frac{e^{1.5}}{4} \).

Key Concepts

Natural Logarithm

In mathematical terms, the natural logarithm is a logarithm with the base of Euler's number, denoted as \( e \). This number is approximately equal to 2.71828. The natural logarithm of a number \( x \), written as \( \ln(x) \), answers the question: "To what power must \( e \) be raised, to produce the number \( x \)?"
For example, if \( \ln(y) = z \), then it implies that \( e^z = y \). This concept becomes particularly useful in solving equations involving growth and decay processes, like exponential growth in populations or radioactive decay in physics. It's crucial to understand that the natural logarithm is the inverse operation of exponentiation, similar to how subtraction is the inverse of addition.

Exponentiation

Exponentiation involves raising a number, called the base, to the power of an exponent. The expression \( e^{1.5} \), which appears in the exercise, is an instance of exponentiation, where the base \( e \) is raised to the power of 1.5.
In the context of solving logarithmic equations, exponentiation is utilized to "cancel out" a logarithm. This is based on the principle that exponentiating the natural logarithm returns the original number. For example, if we have a natural logarithm \( \ln(a) \), it can be rewritten using exponentiation as \( a = e^{\ln(a)} \).
Thus, in the exercise, exponentiating both sides of \( \ln(4x) = 1.5 \) provides us with \( 4x = e^{1.5} \), effectively eliminating the logarithm and paving the way for isolating \( x \) in the equation.

Exact Solutions

Exact solutions in mathematics refer to answers that are presented in their most precise terms, without approximations. In equations like the one given in the exercise, exact solutions use mathematical constants and expressions, such as \( e^{1.5} \) rather than an approximate decimal.
For solving \( \ln(4x) = 1.5 \), the process of finding an exact solution involved isolating \( x \) to find \( x = \frac{e^{1.5}}{4} \).

• An exact solution allows for a more exact interpretation, particularly useful in contexts where precision is critical, such as engineering or theoretical physics.
• This solution maintains the mathematical integrity without losing accuracy that decimal approximations might introduce.

Calculators or computational tools might be used to verify such solutions numerically, but the fundamental exact form is preferred for theoretical analysis.

Mathematical Problem Solving

Mathematical problem solving involves applying various techniques and logical steps to find solutions to given problems.
In the context of logarithmic equations, it's important to first understand the nature of the problem, which involves isolating the variable, transforming the equation if necessary, and using mathematical rules to find the solution.
In the step-by-step solution of the exercise \( \ln(4x) = 1.5 \), these steps were:

• Understand the problem and decide on the approach needed.
• Isolate the logarithmic expression, if necessary.
• Use exponentiation to eliminate the logarithm.
• Finally, solve for the variable and simplify if needed.

These systematic steps are fundamental in honing problem-solving skills, not just in mathematics, but across various disciplines, equipping learners with the skills to tackle complex problems.