Problem 7 - Calculus

Question

Solve for \(x\) using logs. $$3^{x}=11$$

Step-by-Step Solution

Answer

The solution is approximately \(x \approx 2.1827\).

Step 1: Apply the Logarithm

To solve the equation \(3^x = 11\), we first apply a logarithm to each side. A common choice is the natural logarithm (ln). We have: \(\ln(3^x) = \ln(11)\).

Step 2: Use the Power Rule of Logarithms

Apply the power rule of logarithms which states \(\ln(a^b) = b \cdot \ln(a)\). Thus, we simplify the left side: \(x \cdot \ln(3) = \ln(11)\).

Step 3: Isolate the Variable

To solve for \(x\), divide both sides of the equation by \(\ln(3)\) to isolate \(x\): \(x = \frac{\ln(11)}{\ln(3)}\).

Step 4: Calculate the Result

Use a calculator to compute \(\ln(11)\) and \(\ln(3)\). Then, divide the two results to find \(x\). This gives us approximately: \(x \approx \frac{2.3979}{1.0986} \approx 2.1827\).

Key Concepts

Exponential Equations

Exponential equations, like \(3^x = 11\) in this example, have variables in the exponent position. These types of equations are frequently encountered in algebra and can initially seem challenging. However, tackling them becomes straightforward with the right approach, often involving logarithms.

The primary goal is to find the value of the variable that makes the equation true. In an exponential equation, the expression on one side is a power of a base. Here, \(3\) is raised to the power \(x\) and needs to equal \(11\). This scenario requires reversing the action of exponentiation.

Often, logarithms serve this purpose, transforming the equation into a form where the variable can be isolated. If the equation is not easily solvable using simple mental math, applying a logarithmic function allows us to unravel the exponent and solve for the variable.

Natural Logarithm

The natural logarithm is a specific logarithmic function denoted by \(\ln\), where the base is the irrational number \(e \approx 2.71828\). This is an essential mathematical constant that emerges in continuous compounding and growth rate scenarios.

In the context of our exercise, \(\ln(3^x) = \ln(11)\), we apply the natural logarithm to both sides of the exponential equation. Doing so brings the unknown variable down from the exponent, making it possible to solve the equation.

The choice of using natural logarithms often depends on the functions encountered in calculus and natural growth models, though common logarithms (base 10) could also be used. Regardless of the type, the key function of any logarithm in this context is to simplify the manipulation of exponential expressions.

Power Rule of Logarithms

The power rule of logarithms is a fundamental property that simplifies expressions where a logarithm is raised to a power. For any positive numbers \(a\) and \(b\), and \(a eq 1\), the rule states \(\ln(a^b) = b \cdot \ln(a)\). This means you can "bring down" the exponent \(b\) as a coefficient.

In solving \(3^x = 11\), the rule allows the transformation of \(\ln(3^x)\) into \(x \cdot \ln(3)\). The power rule thus makes it easier to isolate \(x\), converting the equation into a linear form.

After simplifying using the power rule, as seen in the solution, the equation becomes \(x \cdot \ln(3) = \ln(11)\). This straightforward linear equation can then be easily manipulated to solve for \(x\), illustrating the power rule's ability to simplify calculations involving exponents.