Problem 57 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, solve for \(x\) or \(t\). $$ 3 \ln 5 x=8 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{e^{8/3}}{5}\).

Step 1: Isolate the Logarithm

In the equation \(3 \ln 5 x = 8\), begin by isolating the logarithm on one side of the equation. This can be done by dividing both sides of the equation by 3. This gives us the equation \(\ln 5x = \frac{8}{3}\).

Step 2: Rewrite in Exponential Form

Next, we want to get rid of the natural logarithm (ln). We can do this by converting the equation into its equivalent exponential form. Since the base of ln is \(e\), this gives us the equation \(e^{8/3} = 5x\).

Step 3: Solve for x

Finally, to solve for \(x\), we need to isolate \(x\) on one side of the equation. To accomplish this, divide both sides of the equation by 5. This results in the final solution \(x = \frac{e^{8/3}}{5}\).

Key Concepts

Exponential Equations

Exponential equations are equations in which variables appear in exponents. These types of equations can often be intimidating because they involve concepts of powers and roots. However, they can be simplified using various techniques, such as converting logarithmic equations into exponential form.When converting from a logarithmic equation to an exponential one, remember that the logarithm base is the same as the base of the exponent. For example, in the equation where the natural logarithm is involved, the base is the mathematical constant \(e\), approximately equal to 2.718. This means any expression like \(\ln(x) = y\) can be rewritten as the exponential equation \(e^y = x\).The concept of exponential equations is vital as it allows us to go between different mathematical representations, making it easier to solve for unknowns.

Natural Logarithms

Natural logarithms, often written as \(\ln x\), use the base \(e\). The natural logarithm is particularly useful in solving equations where variables are in exponents or involve exponential growth models.A logarithm is essentially the inverse of exponentiation. That means if you have an equation such as \(\ln 5x = \frac{8}{3}\), this can be transformed into an exponential form for further simplification. This transformation leverages the fact that the natural logarithm with base \(e\) leads to the expression \(e^{8/3} = 5x\).Natural logarithms simplify many complex calculations into manageable steps, often making them the tool of choice in calculus and compound interest problems. That's why understanding how to manipulate and convert these equations is essential for proficiency in mathematics.

Isolation of Variables

The goal in many algebraic problems, like the one presented here, is to solve for a specific variable. This process is known as the isolation of variables. Isolation involves manipulating the equation until the variable of interest stands alone on one side of the equation.In the provided exercise, the isolation process began by dividing the entire equation by 3 to handle the coefficient in front of the natural logarithm: from \(3 \ln 5x = 8\) to \(\ln 5x = \frac{8}{3}\). This is a clear example of isolating the logarithmic expression as a prelim step.Ultimately, the process concludes by isolating \(x\) in the expression \(5x = e^{8/3}\). Further division by 5 results in \(x = \frac{e^{8/3}}{5}\), successfully isolating \(x\). Mastery of this technique allows one to systematically address and resolve equations across a variety of mathematical problems.