Problem 4 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, rounded to four decimal places. \(10^{-x}=4\)

Step-by-Step Solution

Answer

x = -0.6021

Step 1: Rewrite the Equation Using Logarithms

To solve for \(x\), we first rewrite the equation using logarithms. We have the equation \(10^{-x} = 4\). We will take the logarithm of both sides. Since we are dealing with base 10, we will use the common logarithm (log base 10). This gives:\[\log(10^{-x}) = \log(4)\]

Step 2: Use Logarithmic Properties

Now, use the property of logarithms that \(\log(a^b) = b \cdot \log(a)\). Therefore, \(\log(10^{-x}) = -x \cdot \log(10)\). Since \(\log(10) = 1\), it simplifies to:\[-x = \log(4)\]

Step 3: Solve for -x

Now, solve for \(-x\) by evaluating the logarithm:\[-x = \log(4) \approx 0.6021\]

Step 4: Solve for x

Since \(-x = 0.6021\), to solve for \(x\), we multiply both sides of the equation by -1:\[x = -0.6021\]

Step 5: Round to Four Decimal Places

Finally, ensure the answer is rounded to four decimal places. In this case, \(x = -0.6021\) is already in the required format.

Key Concepts

Logarithms

Logarithms are a mathematical concept that help us understand the relationship between numbers in exponential form. Essentially, a logarithm answers the question: "To what power must the base be raised, to produce a given number?". For example, in the equation \(\log_b(a) = c\), the base \(b\) raised to the power \(c\) gives the number \(a\). This is why logarithms are often described as the inverse of exponents.

When working with logarithms, it is crucial to understand the different types. The most common ones are common logarithms (base 10) and natural logarithms (base \(e\)). In the exercise with an exponential equation \(10^{-x} = 4\), we use common logarithms because the equation involves base 10.

This method simplifies the process, allowing us to express exponential equations in a more straightforward linear form. Thus, it becomes much easier to solve for the unknown variable.

Logarithmic Properties

Understanding logarithmic properties is key to simplifying and solving equations involving logarithms. Some important properties include:

• Product Rule: \(\log_b(xy) = \log_b(x) + \log_b(y)\)
• Quotient Rule: \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\)
• Power Rule: \(\log_b(x^a) = a\log_b(x)\)
• Change of Base Formula: \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\) for any positive base \(c\)

In our exercise, we primarily use the Power Rule. When we take the logarithm of both sides of the equation \(10^{-x} = 4\), we apply the Power Rule to express the left side as \(-x\times\log(10)\). This simplifies our equation significantly, making it easier to isolate and solve for \(x\).

The properties of logarithms are powerful tools. They enable us to handle exponential equations more gracefully by breaking them into fundamental components.

Solving Equations

Solving equations, especially those involving exponents, can initially seem complex. However, by using logarithms, we can transform them into simpler forms. Let's go through the steps that were used in the provided solution.

The first step involved recognizing that our equation \(10^{-x} = 4\) can be rewritten in terms of common logarithms. By doing so, we set up a relationship that requires finding the logarithmic value of both sides: \(\log(10^{-x}) = \log(4)\). This approach directly employs the inverse relationship between logarithms and exponents.

Next, the application of logarithmic properties allows further simplification. Using the property that \(\log(a^b) = b \cdot \log(a)\), the equation transforms into \(-x \cdot \log(10) = \log(4)\). Since \(\log(10)\) equals 1, this simplifies to \(-x = \log(4)\).

The last part involves straightforward algebra to solve for \(x\). By multiplying both sides by \(-1\), we solve for \(x\) and obtain \(x = -0.6021\). This demonstrates how utilizing logarithms and their properties effectively reduces the complexity involved in solving exponential equations.