Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(6^{5 x}=3000\)

Step-by-Step Solution

Answer

The solution to the equation is \(x \approx 0.579\).

Step 1: Take the logarithm of both sides

Use the logarithmic form to convert the exponential equation. Both sides of the equation are logged to base 10:\(\log(6^{5x}) = \log(3000)\)

Step 2: Use the properties of logarithms

Apply the power rule of logarithms, which allows to move the exponent of the argument to the front of the logarithm:\(5x \cdot \log(6) = \log(3000)\)

Step 3: Solve for the variable

Rearrange the equation to solve for \(x\):\(\dfrac{\log(3000)}{\log(6)} \cdot \dfrac{1}{5}= x\)

Step 4: Approximate the value

Calculate the value of \(x\) and approximate it to three decimal places:\(x \approx 0.579\).

Key Concepts

Logarithm Properties

Understanding the properties of logarithms is crucial when solving exponential equations. A logarithm is essentially an exponent to which a base must be raised to produce a certain number. For example, if you have the equation \( b^y = x \), you can express it in logarithmic form as \( \log_b(x) = y \).

One of the key properties we use in solving exponential equations is the Power Rule, which states that \( \log_b(m^n) = n \cdot \log_b(m) \). This property allows us to 'bring down' the exponent, making the equation much easier to handle algebraically. Another important property is the Change of Base Formula, which lets us convert a logarithm to a different base if needed: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \), where \(c\) is any positive value.

These properties are not only helpful in solving equations but also enable us to understand the behavior and relationships between exponential expressions and their logarithmic counterparts.

Algebraic Solution

Finding an algebraic solution to an equation involves manipulating the equation using algebraic rules until the variable of interest is isolated. For exponential equations, this often includes using properties of logarithms, as seen in the step-by-step solution provided. Once the equation is in logarithmic form, it becomes a matter of using basic algebraic techniques.

For example, after applying the power rule, we have a simple equation to solve for \(x\). This involves rearranging the terms and isolating \(x\) on one side of the equal sign. Such algebraic manipulation is fundamental in all areas of mathematics and is particularly useful when equations become too complex for simple inspection. By breaking down the process into manageable steps and systematically applying algebraic principles, we can derive a clear path to the solution.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are represented in the form of \( b^x \) and are notable for their unique features, such as rapid growth or decay. These functions appear in many real-world applications, including compound interest calculations, population growth models, and radioactive decay.

When solving equations involving exponential functions, it’s vital to recognize that logarithms are the inverse operations of exponentiation. This relationship is why we can apply logarithms to both sides of an exponential equation to 'unravel' the exponent and solve for the variable. By mastering exponential functions and their properties, students gain a powerful tool for solving a variety of practical and theoretical problems in mathematics and science.