Problem 79
Question
Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship in a coordinate system whose axes are accordingly transformed so that a straight line results. $$ y=2^{x} ; \text { base } 2 $$
Step-by-Step Solution
Verified Answer
Transform \( y=2^x \) into \( \log_2(y) = x \); graph it as a straight line \( y = x \) in transformed axes.
1Step 1: Understand the exponential relationship
The given exponential function is \( y = 2^x \). This means for every unit increase in \( x \), \( y \) is multiplied by 2.
2Step 2: Apply logarithmic transformation
To linearize the relationship, apply the logarithm with base 2 to both sides of the equation. This gives \( \log_2(y) = \log_2(2^x) \).
3Step 3: Use properties of logarithms
According to the property \( \log_b(b^a) = a \), we simplify \( \log_2(2^x) \) to \( x \). Thus, \( \log_2(y) = x \) is the linear relationship.
4Step 4: Graph the linear relationship
In a coordinate system where the x-axis is \( \log_2(y) \) and the y-axis is \( x \), the equation \( \log_2(y) = x \) becomes a straight line through the origin with a slope of 1.
Key Concepts
Exponential FunctionLinear RelationshipProperties of Logarithms
Exponential Function
Exponential functions play a crucial role in various fields, showing how quantities grow or decay at consistent rates. In the given exercise, the exponential function is defined as \( y = 2^x \). Here, each time \( x \) increases by 1, \( y \) doubles. This is a unique characteristic of exponential growth, where the base number, in this case, 2, indicates the growth factor.
Exponential functions are not limited to growth; they can also represent decay, depending on the base. If the base is a fraction between 0 and 1, the function illustrates exponential decay. This transformation to an exponential form is crucial for understanding phenomena like population growth, radioactive decay, and interest calculations.
Exponential functions are characterized by their curved graphs, representing rapid increases or decreases rather than fixed changes. This curvature makes them distinct from linear graphs, prompting the need for logarithmic transformations to analyze and compare growth rates in a more straightforward manner.
Exponential functions are not limited to growth; they can also represent decay, depending on the base. If the base is a fraction between 0 and 1, the function illustrates exponential decay. This transformation to an exponential form is crucial for understanding phenomena like population growth, radioactive decay, and interest calculations.
Exponential functions are characterized by their curved graphs, representing rapid increases or decreases rather than fixed changes. This curvature makes them distinct from linear graphs, prompting the need for logarithmic transformations to analyze and compare growth rates in a more straightforward manner.
Linear Relationship
Linear relationships are the simplest type of mathematical relationship, represented by a straight line on a graph. Upon applying the logarithmic transformation to the exponential equation \( y = 2^x \), we obtain a linear equation: \( \log_2(y) = x \).
In a linear equation, there's a constant rate of change, indicated by a constant slope. The linear relationship \( \log_2(y) = x \) suggests that for every unit increase in \( x \), \( \log_2(y) \) increases by the same amount—hence, a straight line when plotted.
Linear relationships are significant because they simplify complex problems, making them easier to solve and understand. For instance, in physical sciences, linear relationships help to predict outcomes and analyze trends, providing insights into the consistency or rate of change within a system. This exercise shows the transformation of a curved exponential graph into a straight line, facilitating easier analysis and interpretation.
In a linear equation, there's a constant rate of change, indicated by a constant slope. The linear relationship \( \log_2(y) = x \) suggests that for every unit increase in \( x \), \( \log_2(y) \) increases by the same amount—hence, a straight line when plotted.
Linear relationships are significant because they simplify complex problems, making them easier to solve and understand. For instance, in physical sciences, linear relationships help to predict outcomes and analyze trends, providing insights into the consistency or rate of change within a system. This exercise shows the transformation of a curved exponential graph into a straight line, facilitating easier analysis and interpretation.
Properties of Logarithms
Understanding the properties of logarithms is key to unraveling exponential and logarithmic functions. Logarithms allow us to linearize exponential functions, making them easier to handle mathematically. In this example, the property \( \log_b(b^a) = a \) helps us simplify the expression \( \log_2(2^x) \) to \( x \). This simplification is possible because logarithms and exponents are inverse operations.
Logarithms have other useful properties, such as the product rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \), and the quotient rule: \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \). These properties are essential in solving complex equations involving exponential relationships by breaking them into more manageable parts.
By applying logarithmic transformations, we convert the curved nature of exponential functions into linear ones. This transformation simplifies graph plotting and enhances our ability to predict and interpret data. The smoothing effect of logarithms on exponential graphs is particularly vital in scientific and financial applications, where precision and clarity are paramount.
Logarithms have other useful properties, such as the product rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \), and the quotient rule: \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \). These properties are essential in solving complex equations involving exponential relationships by breaking them into more manageable parts.
By applying logarithmic transformations, we convert the curved nature of exponential functions into linear ones. This transformation simplifies graph plotting and enhances our ability to predict and interpret data. The smoothing effect of logarithms on exponential graphs is particularly vital in scientific and financial applications, where precision and clarity are paramount.
Other exercises in this chapter
Problem 78
Find the inverse of \(f(x)=\left(\frac{1}{3}\right)^{x}, x \in \mathbf{R}\), together with its domain, and graph both functions in the same coordinate system.
View solution Problem 78
Which real number \(x\) satisfies (a) \(\log _{4} 64=x ?\) (b) \(\log _{1 / 5} 625=x ?\) (c) \(\log _{10} 10,000=x\) ?
View solution Problem 79
Find the inverse of \(f(x)=2^{x}, x \geq 0\), together with its domain, and graph both functions in the same coordinate system.
View solution Problem 79
Simplify the following expressions: (a) \(-\ln \frac{1}{3}\) (b) \(\log _{4}\left(x^{2}-4\right)\) (c) \(\log _{2} 4^{3 x-1}\)
View solution