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Magazine Chapter 2: Problem 3
Simplify each of the following to a single log term:
(a) \(3 \ln t-\ln t\)
(b) \(6 \log t^{2}+4 \log t\)
(c) \(\ln \left(3 y^{6}\right)-2 \ln 3+\ln y\)
(d) \(\ln (6 x+4)-\ln (3 x+2)\)
(e) \(\frac{\log (9 x)}{2}-\log \left(\frac{2}{3 x}\right)\)
4 Sketch graphs of the following functions, using the same axes:
$$
y=\ln (2 x), \quad y=\ln x \quad 0
Short Answer
Expert verified
(a) \(2 \ln t\); (b) \(\log t^{16}\); (c) \(\ln \left(\frac{y^{7}}{3}\right)\); (d) \(\ln \left(\frac{6x+4}{3x+2}\right)\); (e) \(\log \left(\frac{9x^{3/2}}{2}\right)\); Vertical distance between graphs is \(\ln 2\).
Step by step solution
01
Simplify expression (a)
The expression is given as \(3 \ln t - \ln t\). Here, \( \ln t\) can be factored out as a common factor. \[ 3 \ln t - \ln t = (3 - 1) \ln t = 2 \ln t \]
02
Simplify expression (b)
The expression is \(6 \log t^{2} + 4 \log t\). Using the power rule for logarithms \(a \log b = \log b^{a} \), rewrite the expression:\[ 6 \log t^{2} = \log (t^{2})^{6} = \log t^{12} \]Add this to the transformed version of the second term:\[ 4 \log t = \log t^{4} \]So, the expression becomes:\[ \log t^{12} + \log t^{4} = \log (t^{12} \cdot t^{4}) = \log t^{16} \]
03
Simplify expression (c)
The expression is given as \( \ln(3y^6) - 2\ln3 + \ln y \). Apply the property of logs that allows you to split or combine logs:\[ \ln(3y^6) = \ln 3 + 6\ln y \]Substitute into the expression:\[ \ln 3 + 6\ln y - 2\ln 3 + \ln y \]Combine like terms:\[ (\ln 3 - 2\ln 3) + (6\ln y + \ln y) = -\ln 3 + 7\ln y = \ln y^{7} - \ln 3 = \ln \left( \frac{y^{7}}{3} \right) \]
04
Simplify expression (d)
The expression is \( \ln (6x+4) - \ln (3x+2) \). Use the quotient rule for logarithms \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \):\[ \ln \left( \frac{6x+4}{3x+2} \right) \]
05
Simplify expression (e)
The expression is given as \( \frac{\log (9x)}{2} - \log \left(\frac{2}{3x}\right) \). First, apply the power rule on the first term:\[ \frac{1}{2}\log (9x) = \log (9x)^{1/2} = \log \sqrt{9x} = \log (3\sqrt{x}) \]Then rewrite the second term using the property \( \log \left( \frac{a}{b} \right) = \log a - \log b \):\[ \log \left( \frac{2}{3x} \right) = \log 2 - \log (3x) \]Thus the expression becomes:\[ \log (3\sqrt{x}) - (\log 2 - \log (3x)) \]Combine them:\[ \log (3\sqrt{x}) + \log (3x) - \log 2 = \log \left( \frac{3x \cdot 3\sqrt{x}}{2} \right) = \log \left( \frac{9x^{3/2}}{2} \right) \]
06
Sketch the graphs of functions
We're given functions \( y=\ln (2x) \) and \( y=\ln x \) for \( 0 < x \leq 10 \). These two functions share the same shape since they are both logarithmic but shifted vertically.1. For \( y=\ln (2x) \), this graph is shifted upwards compared to \( y=\ln x \) by the value \( \ln 2 \).2. The vertical distance between the graphs at any \( x \) is therefore a constant \( \ln 2 \).Measure distances:- At \( x=1 \): \( \ln (2 \cdot 1) - \ln 1 = \ln 2 \)- At \( x=2 \): \( \ln (2 \cdot 2) - \ln 2 = \ln 2 \)- At \( x=8 \): \( \ln (2 \cdot 8) - \ln 8 = \ln 2 \) This confirms that \( \ln (2x) = \ln 2 + \ln x \) explains the consistent vertical distance.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logarithm Properties
Understanding logarithm properties is crucial for simplifying logarithmic expressions and solving logarithmic equations. Let's summarize three main properties of logarithms that are often used in exercises like the one given:
- Product Property: \( \log_b(M \cdot N) = \log_b M + \log_b N \). This allows us to convert the logarithm of a product into a sum of logarithms, which is useful when expanding or combining logarithmic terms.
- Quotient Property: \( \log_b\left( \frac{M}{N} \right) = \log_b M - \log_b N \). This property lets us rewrite the logarithm of a quotient as the difference of logarithms, simplifying the subtraction of log items.
- Power Property: \( \log_b(M^n) = n \log_b M \). It becomes handy when dealing with powers inside a logarithmic function because it allows you to bring exponents as a factor outside the log, simplifying multiplication inside the log argument.
Logarithmic Expressions
Logarithmic expressions can look intimidating, but they're manageable once you break them down using logarithm properties. In the given exercise, expressions are simplified step-by-step to demonstrate how these properties are used:
- Combined Logarithms: For example, in part (b), \( 6 \log t^{2} + 4 \log t \) is simplified by using the power and product properties to create a single term: \( \log t^{16} \). The use of transformations, like changing \( 6 \log t^{2} \) to \( \log t^{12} \), highlights how exponents can be manipulated within the logarithm.
- Simplification Steps: Dealing with expressions like \( \ln(3y^6) - 2\ln3 + \ln y \), where terms are combined and reduced to \( \ln \left( \frac{y^{7}}{3} \right) \), demonstrates that simplification is about recognizing like terms and reducing the complexity of the expression.
Logarithmic Functions
Logarithmic functions have unique characteristics. They are often inverses of exponential functions. Here are key details about them:
- Domain and Range: Logarithmic functions only accept positive input values. This is why you see domains typically set in positive ranges, e.g., for \( y = \ln x \), \( x > 0 \). The range, however, can be all real numbers since the output of a log function can extend infinitely.
- Behavior and Growth: Logarithmic functions grow very slowly. They increase towards infinity as \( x \) increases, but at a decreasing rate. This makes them suitable for modeling scenarios where variables stabilize over time.
- Horizontal Shifts and Transformations: Any multiplication inside the log function, such as \( \ln(2x) \), shifts the graph vertically. Thus, even a base change inside the log due to a coefficient results in a vertical shift, not altering its fundamental shape.
Graphing Logarithms
Graphing logarithms involves understanding their basic shape and how they change with transformations. Here's how to think about logarithmic graphs:
- Basic Shape: The graph of a logarithmic function such as y = ln x has a basic shape that starts from negative infinity as x approaches zero and passes through the point (1,0), continuing to rise slowly.
- Graph Transformation: When we graph \( y = \ln (2x) \) next to \( y = \ln x \), both graphs are similar in shape, with the graph of \( y = \ln (2x) \) shifted vertically upwards by a constant \( \ln 2 \). This constant vertical shift is because \( \ln(2x) = \ln 2 + \ln x \).
- Comparative Analysis: By measuring the vertical distance between these two graphs at any \( x \) point, we find it always equals \( \ln 2 \). This illustrates logarithmic laws applied spatially and visually, offering a tangible way to understand theoretical properties.
