Problem 81 - Precalculus : Building Concepts and Connections

Question

Solve each equation graphically and express the solution as an appropriate logarithm to four decimal places. If a solution does not exist, explain why. $$4\left(10^{x}\right)=20$$

Step-by-Step Solution

Answer

\nThe approximate solution for the equation \(4\left(10^{x}\right)=20\) is \(x \approx 0.6989\).

Step 1: Rearrange the equation

Let's structure the equation in a form that allows us to use logarithmic properties. We divide both sides by 4, getting \(10^{x} = 5\) .

Step 2: Convert to logarithmic form

To solve for \(x\), conversion of the equation from exponential to logarithmic form is required. Logarithmic form of the equation is \( \log_{10}(5) = x \) .

Step 3: Solve the logarithm

Finally, to get the value for \(x\), we calculate the logarithm. This can be done using a calculator. It provides the answer: \(x \approx 0.6989\).

Key Concepts

Graphical Solutions of Equations

Understanding how to solve equations graphically is an essential skill in mathematics. In essence, it involves plotting the graphs of two expressions and identifying the points of intersection. In the context of solving an exponential equation such as \(4(10^{x})=20\), one would typically plot the functions \(y=4(10^{x})\) and \(y=20\) on the same set of axes.

The point where both graphs intersect represents the solution. It is the value of 'x' for which both equations yield the same 'y' value. However, graphical solutions are not always precise, especially when solutions require a high level of accuracy. Therefore, specifying the solution to four decimal places might necessitate the use of a calculator after getting an approximate value from the graph.

When graphically depicted, if the lines do not intersect, it indicates that there is no real solution to the equation. But when they do, you can express the solution in logarithmic form to achieve the required precision.

Logarithmic Forms

Exponential equations are often solved more easily when converted into logarithmic form. The logarithmic form is based on the principle that if \(a^{b}=c\), then \(b = \text{log}_{a}(c)\). For the given equation \(4(10^{x}) = 20\), after dividing by 4, we have \(10^{x} = 5\).

To convert this into logarithmic form, we apply the definition of a logarithm, yielding \( \text{log}_{10}(5) = x \). This form allows for easier computation, particularly when dealing with unfamiliar exponentials, as it brings the variable 'x' from an exponent to a position where it can be calculated directly, often with the help of a calculator. Understanding how to convert between exponential and logarithmic forms is fundamental to solving equations involving exponentiation and for performing various operations involved in further mathematical analysis.

Logarithmic Properties

Logarithmic properties are rules that govern the operations with logarithms. These properties help simplify complex expressions and solve logarithmic equations. Some fundamental properties include the Product Rule, the Quotient Rule, and the Power Rule.

For instance, the Power Rule states that \( \text{log}_{a}(b^{c}) = c \times \text{log}_{a}(b) \), which is particularly useful when you have an exponent in a logarithm. Similarly, if you need to multiply two logarithms with the same base, the Product Rule can be applied: \( \text{log}_{a}(bc) = \text{log}_{a}(b) + \text{log}_{a}(c) \).

Exploiting these properties can greatly facilitate the simplification of logarithmic expressions and the solving of equations. When it comes to exercises like the one provided, these properties will allow us to understand why logarithmic form is a potent alternative to represent solutions, giving clear insight into the nature of exponential relationships.