Problem 19 - Intermediate Algebra

Question

Complete each solution. Solve: \(\quad 2^{x}=7\) $$ \begin{aligned} _2^{x} &=\log 7 \\ x _=\log 7 \\ x &=\frac{\log 7}{\log 2} \\ x & \approx \end{aligned} $$

Step-by-Step Solution

Answer

\(x \approx 2.807\)

Step 1: Understanding the Problem

We need to solve the equation \(2^x = 7\) for \(x\). This means we need to find the value of \(x\) that makes this equation true.

Step 2: Applying Logarithms

To solve for \(x\), we take the logarithm of both sides of the equation. This allows us to bring the exponent down as a coefficient: \(\log(2^x) = \log(7)\).

Step 3: Using the Power Rule of Logarithms

The power rule of logarithms states that \(\log(a^b) = b \cdot \log(a)\). Applying this to \(\log(2^x)\), we get \(x \cdot \log(2) = \log(7)\).

Step 4: Solving for x

Now, solve for \(x\) by dividing both sides of the equation by \(\log(2)\), resulting in \(x = \frac{\log(7)}{\log(2)}\).

Step 5: Calculating x Numerically

Use a calculator to evaluate \(x \approx \frac{\log(7)}{\log(2)}\). Logarithmic values are approximately: \(\log(7) \approx 0.845\) and \(\log(2) \approx 0.301\). Therefore, \(x \approx \frac{0.845}{0.301} \approx 2.807\).

Key Concepts

Understanding Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In the exercise, the equation \(2^x = 7\) exemplifies an exponential function, with the base being 2 and the exponent being \(x\). Exponential functions have unique characteristics:

• The base, which is typically a positive number, indicates the rate of growth or decay.
• The exponent, which here is a variable, shows how many times the base is multiplied by itself.

Exponential functions are crucial because they represent growth or decay processes in fields like finance, sciences, and engineering. When solving an equation like \(2^x = 7\), you're essentially finding the exponent that makes the equation true, revealing the power to which the base must be raised to produce a particular result.

Applying the Power Rule of Logarithms

The power rule of logarithms is a nifty tool that helps simplify expressions where you have exponents inside a logarithm. The rule states: \(\log(a^b) = b \cdot \log(a)\). This means you can take the exponent out as a multiplier. In the context of our problem \(\log(2^x)\), applying the power rule makes it \(x \cdot \log(2)\). This step is key because it converts the exponent from its position on the base, making it easier to isolate and solve for \(x\). Using this rule simplifies solving equations involving exponents and logarithms, particularly where finding the exact exponent is required, such as in exponential equations. The power rule is widely used in calculus, algebra, and other areas requiring precise calculations and simplifications.

Conquering Logarithmic Calculations

Logarithmic calculations are the backbone of solving exponential equations. The logarithm turns multiplications into additions, which makes calculations simpler and is key in reversing exponential functions.When you have an equation like \(2^x = 7\), logarithms can unify the equation. By taking the log of both sides, \(\log(2^x) = \log(7)\), you leverage the properties that allow separating the exponent from its base through the power rule.For solving, once we have \(x \cdot \log(2) = \log(7)\), divide both sides by \(\log(2)\) to isolate \(x\). Thus, \(x = \frac{\log(7)}{\log(2)}\). Knowing how to calculate these logarithmic values is essential for reaching a solution. Most calculators directly provide logarithms' estimates, helping to finalize the solution. In practice, understanding these calculations is fundamental to mastering problems involving exponential functions and logarithms.