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Chapter 4: Problem 114

Graph \(y=\log x, y=\log (10 x),\) and \(y=\log (0.1 x)\) in the same viewing rectangle. Describe the relationship among the three graphs. What logarithmic property accounts for this relationship?

Short Answer

Expert verified
The relationship between these graphs is explained by the logarithmic relationship where the multiplication inside a logarithm turns into addition, shifting the graph left or right. When the coefficient of \(x\) is multiplied by 10, the graph for \(y = \log (10x)\) shifts left, and when divided by 10, the graph for \(y = \log (0.1x)\) shifts right.

Step by step solution

01

Graphing the First Logarithm

Start by graphing \(y = \log x\), the simplest logarithmic function. The graph will start from (1,0) and increase as \(x\) moves away from 1, demonstrating the property of logarithms where \(\log_b a = n\) only if \(b^n = a\). Thus for base 10, when \(x = 1\), \(y = 0\) because \(10^0 = 1\).
02

Graphing the Second Logarithm

Next, graph the function \(y = \log (10x)\). This graph will show a shift to the left compared to \(y = \log x\), because the coefficient of \(x\) is greater than 1, effectively dividing the \(x\)-values by 10. This demonstrates one of the key properties of logarithms, that \(\log_b (ac) = \log_b a + \log_b c\). When \(x = 0.1\), \(y = 0\), because \(10 \times 0.1 = 1\) and \(\log 1 = 0\).
03

Graphing the Third Logarithm

Finally, graph \(y = \log (0.1x)\). With a smaller \(x\) coefficient, this graph shifts to the right of \(y = \log x\), effectively multiplying the \(x\)-values by 10 as compared to \(y = \log x\). When \(x = 10\), \(y = 0\) since \(0.1 \times 10 = 1\) and \(\log 1 = 0\).
04

Analyzing the Relationship

Analyzing these three graphs, it's clear that changing the coefficient of \(x\) in a logarithm shifts the graph horizontally. Greater coefficients shift the graph towards the left, and smaller ones towards the right. As the coefficient multiplies or divides \(x\), \(\log (ax)\) can be written as \(\log a + \log x\), according to properties of logarithms. This relation results in the shifting of these graphs.

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Most popular questions from this chapter

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the function to solve. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age \(13 ?\)

The bar graph indicates that the percentage of first-year college students expressing antifeminist views declined after \(1970 .\) Use this information to solve. (GRAPH CANNOT COPY). The function $$f(x)=-7.52 \ln x+53$$ models the percentage of first-year college men, \(f(x)\) expressing antifeminist views (by agreeing with the statement) \(x\) years after 1969. a. Use the function to find the percentage of first-year college men expressing antifeminist views in 2008 . Round to one decimal place. Does this function value overestimate or underestimate the percentage displayed by the graph? By how much? b. Use the function to project the percentage of first-year college men who will express antifeminist views in 2015 . Round to one decimal place.

Exercises \(150-152\) will help you prepare for the material covered in the next section. The formula \(A=10 e^{-0.003 t}\) models the population of Hungary, \(A,\) in millions, \(t\) years after 2006 a. Find Hungary's population, in millions, for \(2006,2007\), \(2008,\) and \(2009 .\) Round to two decimal places b. Is Hungary's population increasing or decreasing?

Without using a calculator, determine which is the greater number: \(\log _{4} 60\) or \(\log _{3} 40\).

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the function to solve. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age ten?
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