Problem 14 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, rounded to four decimal places. \(2^{3 x}=34\)

Step-by-Step Solution

Answer

The solution is approximately \(x = 1.6967\).

Step 1: Take the Logarithm of Both Sides

To solve the equation \(2^{3x} = 34\), start by taking the logarithm of both sides. Let's use the natural logarithm (\(\ln\)): \[\ln(2^{3x}) = \ln(34)\] We use logarithms to bring the exponent down as a coefficient.

Step 2: Apply the Power Rule of Logarithms

Apply the power rule \(\ln(a^b) = b \ln(a)\) to the left side of the equation: \[3x \cdot \ln(2) = \ln(34)\] This allows us to solve for \(x\) more easily.

Step 3: Isolate the Variable x

Now, solve for \(x\) by dividing both sides by \(3 \ln(2)\): \[x = \frac{\ln(34)}{3 \ln(2)}\] This isolates \(x\) so we can calculate its value.

Step 4: Calculate the Value

Use a calculator to compute the values: 1. \(\ln(34) \approx 3.5264\)2. \(\ln(2) \approx 0.6931\)3. Substitute into the equation: \[x = \frac{3.5264}{3 \times 0.6931} \approx \frac{3.5264}{2.0793} \approx 1.6967\]Therefore, \(x \approx 1.6967\), rounded to four decimal places.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm with the base of Euler's number, \( e \). The value of \( e \) is approximately 2.71828, and it commonly appears in mathematical applications involving growth and decay processes. When applied to exponential equations, the natural logarithm helps simplify expressions by transforming powers onto coefficients, thereby making equations easier to solve.

For instance, in the equation \(2^{3x}=34\), by taking the natural logarithm of both sides, we can simplify the power expression \(2^{3x}\) into a linear one. This is because logarithms are inherently inverse functions to exponentials, meaning they "undo" the exponential process and allow us to work with the actual exponent directly.

You'll find the natural logarithm very useful in calculus, too, where it is a key function for integration and differentiation, especially when dealing with exponential functions.

Power Rule of Logarithms

The power rule of logarithms is a fundamental property that simplifies the handling of exponents within log expressions. It states that \( \ln(a^b) = b \cdot \ln(a) \). This rule allows us to take an exponent (\( b \)) and move it to the front as a coefficient. This property is vital for solving equations that feature exponential terms.

In our example with \(\ln(2^{3x}) = \ln(34)\), we apply the power rule to change the term on the left to \(3x \cdot \ln(2)\). Here, the exponent \(3x\) becomes a coefficient which can be easily manipulated algebraically. This transformation is crucial in the process of solving the equation as it simplifies the problem into a linear form, where isolating the variable becomes straightforward.

Solving for Variables

Solving for a variable in algebra often revolves around isolating the variable on one side of the equation. This means making sure that the variable you are solving for, in this case \(x\), is alone on one side while everything else is on the opposite side.

In our exercise, after applying the logarithms and the power rule, we simplify the expression to: \(3x \cdot \ln(2) = \ln(34)\). To isolate \(x\), divide both sides by \(3 \cdot \ln(2)\), resulting in \(x = \frac{\ln(34)}{3 \cdot \ln(2)}\).

This isolate-step is crucial as it allows us to solve for \(x\) using basic operations. We simply evaluate the numerical expression that remains: compute \(\ln(34)\) and \(\ln(2)\), then perform the division. This approach is general and can be used for any equation where a variable is influencing an expression through multiplication, division, or any other algebraic operations.