Problem 14 - Precalculus Mathematics for Calculus

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 1.6965.

Step 1: Isolate the Exponential Term

The equation we need to solve is given as \(2^{3x} = 34\). The exponential term \(2^{3x}\) is already isolated on one side of the equation, so no additional steps are needed.

Step 2: Take the Logarithm

To solve for \(x\), take the logarithm of both sides. We'll use the logarithm base 10 for simplicity:\[\log(2^{3x}) = \log(34)\]

Step 3: Use Logarithmic Identity

Apply the power rule of logarithms, \(\log(a^b) = b \cdot \log(a)\):\[3x \cdot \log(2) = \log(34)\]

Step 4: Solve for x

Rearrange the equation to solve for \(x\):\[x = \frac{\log(34)}{3 \cdot \log(2)}\] Calculate the numerical value using a calculator:\(x \approx \frac{1.5315}{3 \cdot 0.3010} = \frac{1.5315}{0.9030}\).

Step 5: Calculate and Round

Perform the division:\[x \approx 1.6965\] Round the result to four decimal places as required.

Key Concepts

Logarithmic Functions

Logarithmic functions are powerful tools in mathematics, especially when dealing with exponential equations. Understanding logarithms can simplify solving equations where exponents are involved. In simple terms, a logarithm is the inverse of an exponentiation. If you have an equation in the form of \(a^b = c\), by taking the logarithm, you can find \(b\) by transforming the equation into \(b = \log_a(c)\).

There are different bases of logarithms, but the most common are base 10, often written as \(\log\), and base \(e\), the natural logarithm, denoted as \(\ln\). In many practical cases, especially when using calculators or software, base 10 is preferred for simplicity.

• Logarithms allow the conversion of multiplicative factors into additive ones, simplifying calculations.
• Understanding logarithms can help interpret exponential growth or decay, such as in population models or radioactive decay.
• The notation \(\log_{base}(value)\) is used to represent logarithms, simplifying expressions and providing an easier path for calculating exponents.

To solve exponential equations, logarithms act as a bridge, allowing us to handle large and complex expressions by converting them to more manageable forms.

Power Rule of Logarithms

The power rule of logarithms is an essential property that greatly simplifies problems involving exponents. The rule states that \(\log(a^b) = b \cdot \log(a)\), allowing you to bring down the exponent as a multiplier in front of the logarithm.

This rule is vital when solving problems where an unknown variable is in the exponent, as seen in exponential equations. By applying the power rule, we can transform the problem into a linear one, which is easier to solve.

Here is why it's useful:

• It helps to "unpack" an exponent, making the equation straightforward without complex powers hindering the process.
• You can see the equation in an easier form, thus reducing potential errors in solving.
• It bridges exponential and linear worlds, which is fundamental to understanding and exploring deeper mathematical concepts.

Using the power rule makes solving equations with logarithms quicker and more efficient, transforming complicated exponential expressions into manageable linear equations.

Solving Equations with Logarithms

Solving exponential equations with logarithms is a methodical process that hinges on understanding the relationship between exponents and logarithms. For an equation like \(2^{3x} = 34\), the main objective is to isolate the variable \(x\).

Here’s a simple step-by-step approach to solve exponential equations like these:

1. Ensure the exponential term is isolated on one side of the equation. This might require manipulation, like dividing or multiplying both sides by a constant.
2. Take the logarithm of both sides. It does not matter too much which base you use, as long as you are consistent, but base 10 \(\log\) or natural log \(\ln\) are most common.
3. Apply the power rule of logarithms to "bring down" the exponent, transforming the exponential term into a linear form.
4. Solve for the variable by performing algebraic operations, like division or multiplication.

Once these steps are followed, you can easily compute the value of \(x\) with the help of a calculator, allowing you to manage and simplify even complex exponential equations proficiently.