Problem 71 - College Algebra with Corequisite Support

Question

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(3^{4 x-5}=38\) using the common log

Step-by-Step Solution

Answer

The approximate value of \(x\) is 2.335.

Step 1: Isolate the Exponential Expression

The given equation is \(3^{4x-5} = 38\). The exponential part \(3^{4x-5}\) is already isolated.

Step 2: Take the Common Logarithm

Apply the common logarithm (log base 10) to both sides of the equation to help isolate the variable. This yields: \[\log(3^{4x-5}) = \log(38)\]

Step 3: Apply Logarithmic Identities

Use the power rule of logarithms which states that \(\log(a^b) = b \cdot \log(a)\). Apply this to the left-hand side of the equation:\[(4x - 5) \cdot \log(3) = \log(38)\]

Step 4: Solve for 4x-5

Divide both sides of the equation by \(\log(3)\) to solve for \(4x - 5\):\[4x - 5 = \frac{\log(38)}{\log(3)}\]

Step 5: Solve for x

Add 5 to both sides of the equation to isolate \(4x\): \[4x = \frac{\log(38)}{\log(3)} + 5\]Now, divide both sides by 4 to solve for \(x\):\[x = \frac{\frac{\log(38)}{\log(3)} + 5}{4}\]

Step 6: Calculate the Result

Use a calculator to compute the approximate value of \(x\). 1. Calculate \(\log(38)\) and \(\log(3)\) using a calculator.2. Compute \(\frac{\log(38)}{\log(3)}\).3. Add 5 to the result from step 2.4. Divide by 4 to find the approximate value of \(x\). The result is approximately 2.335.

Key Concepts

Logarithms

Logarithms are an essential mathematical concept that help us solve equations involving exponential expressions. They are the inverse operations of exponentials, meaning they "undo" the process of raising a number to a power. For instance, if you know that \(2^3 = 8\), a logarithm lets you find the power back by saying \(\log_2(8) = 3\). When dealing with exponential equations, logarithms are particularly useful because they allow us to bring down exponents and work with the expression more directly. Instead of manipulating large numbers, we can work with their logarithms, which are often easier to manage. This makes logarithms a powerful tool in algebra, especially for equations involving variables in the exponent.

Power Rule of Logarithms

The power rule of logarithms is a crucial property that facilitates solving equations like the one in the exercise. The power rule states that \(\log(a^b) = b \cdot \log(a)\). This means when you take the logarithm of an exponential expression, you can "move" the exponent in front of the logarithm, transforming it into a multiplication problem. In our example equation \(\log(3^{4x-5}) = \log(38)\), we use the power rule to shift \(4x - 5\) out of the exponent:

• The expression \(3^{4x-5}\) becomes \((4x - 5) \cdot \log(3)\)
• This simplification allows us to handle the exponent as a linear term

Using properties like the power rule helps us turn complex equations into simpler forms, making it easier to isolate the variable and solve the problem.

Common Logarithm

A common logarithm is a logarithm with base 10, typically written as \(\log(x)\), without a base indicated explicitly. It is the most widely used logarithm in calculations because our number system is base 10, making common logarithms natural to work with. When solving equations involving exponentials, the common logarithm is often used since it's conveniently accessible on most scientific calculators.

• In our equation, applying \(\log(3^{4x-5}) = \log(38)\) uses common logs
• This allows us to calculate the value of \(x\) using a standard calculator, utilizing the \(\log\) function

Understanding when and why to use common logarithms helps streamline the solving process, especially during the step of transforming complex exponential expressions into manageable linear ones.

Approximating Solutions

Approximating solutions is a practical aspect of solving equations, especially when the exact solution is not easily attainable or overly complex. In many real-world scenarios, the exact value is less important than a sufficiently accurate estimate, which is where approximation becomes vital. In our problem, after simplifying the equation to \(x = \frac{\frac{\log(38)}{\log(3)} + 5}{4}\), we use a calculator to find numerical approximations for the logarithmic values.

• Calculate \(\log(38)\) and \(\log(3)\) with a calculator
• Use these values to approximate the fraction \(\frac{\log(38)}{\log(3)}\)
• Add 5 and divide by 4 to find \(x\)

This process results in an approximate solution \(x \approx 2.335\). Approximating allows us to quickly and efficiently find usable answers, particularly when a precise solution may not be necessary.