Solutions
Solutions to Try Its
1. [latex]\left(2,\infty \right)[/latex] 2. [latex]\left(5,\infty \right)[/latex] 3. The domain is [latex]\left(0,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is x = 0.![Graph of f(x)=log_(1/5)(x) with labeled points at (1/5, 1) and (1, 0). The y-axis is the asymptote.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25201915/CNX_Precalc_Figure_04_04_0062.jpg)
![Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1, 0), and (3, 1).The translation function f(x)=log_3(x+4) has an asymptote at x=-4 and labeled points at (-3, 0) and (-1, 1).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25201917/CNX_Precalc_Figure_04_04_0092.jpg)
![Graph of two functions. The parent function is y=log_2(x), with an asymptote at x=0 and labeled points at (1, 0), and (2, 1).The translation function f(x)=log_2(x)+2 has an asymptote at x=0 and labeled points at (0.25, 0) and (0.5, 1).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25201918/CNX_Precalc_Figure_04_04_0122.jpg)
![Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=(1/2)log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (16, 1).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25201919/CNX_Precalc_Figure_04_04_0152.jpg)
![Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25201920/CNX_Precalc_Figure_04_04_0172.jpg)
![Graph of f(x)=-log(-x) with an asymptote at x=0.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25201921/CNX_Precalc_Figure_04_04_0202.jpg)
9. [latex]x\approx 3.049[/latex]
10. x = 1 11. [latex]f\left(x\right)=2\mathrm{ln}\left(x+3\right)-1[/latex]Solutions to Odd-Numbered Exercises
1. Since the functions are inverses, their graphs are mirror images about the line y = x. So for every point [latex]\left(a,b\right)[/latex] on the graph of a logarithmic function, there is a corresponding point [latex]\left(b,a\right)[/latex] on the graph of its inverse exponential function. 3. Shifting the function right or left and reflecting the function about the y-axis will affect its domain. 5. No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers. 7. Domain: [latex]\left(-\infty ,\frac{1}{2}\right)[/latex]; Range: [latex]\left(-\infty ,\infty \right)[/latex] 9. Domain: [latex]\left(-\frac{17}{4},\infty \right)[/latex]; Range: [latex]\left(-\infty ,\infty \right)[/latex] 11. Domain: [latex]\left(5,\infty \right)[/latex]; Vertical asymptote: x = 5 13. Domain: [latex]\left(-\frac{1}{3},\infty \right)[/latex]; Vertical asymptote: [latex]x=-\frac{1}{3}[/latex] 15. Domain: [latex]\left(-3,\infty \right)[/latex]; Vertical asymptote: x = –3 17. Domain: [latex]\left(\frac{3}{7},\infty \right)[/latex]; Vertical asymptote: [latex]x=\frac{3}{7}[/latex] ; End behavior: as [latex]x\to {\left(\frac{3}{7}\right)}^{+},f\left(x\right)\to -\infty [/latex] and as [latex]x\to \infty ,f\left(x\right)\to \infty [/latex] 19. Domain: [latex]\left(-3,\infty \right)[/latex] ; Vertical asymptote: x = –3; End behavior: as [latex]x\to -{3}^{+}[/latex] , [latex]f\left(x\right)\to -\infty [/latex] and as [latex]x\to \infty [/latex] , [latex]f\left(x\right)\to \infty [/latex] 21. Domain: [latex]\left(1,\infty \right)[/latex]; Range: [latex]\left(-\infty ,\infty \right)[/latex]; Vertical asymptote: x = 1; x-intercept: [latex]\left(\frac{5}{4},0\right)[/latex]; y-intercept: DNE 23. Domain: [latex]\left(-\infty ,0\right)[/latex]; Range: [latex]\left(-\infty ,\infty \right)[/latex]; Vertical asymptote: x = 0; x-intercept: [latex]\left(-{e}^{2},0\right)[/latex]; y-intercept: DNE 25. Domain: [latex]\left(0,\infty \right)[/latex]; Range: [latex]\left(-\infty ,\infty \right)[/latex]; Vertical asymptote: x = 0; x-intercept: [latex]\left({e}^{3},0\right)[/latex]; y-intercept: DNE 27. B 29. C 31. B 33. C 35.![Graph of two functions, g(x) = log_(1/2)(x) in orange and f(x)=log(x) in blue.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25201923/CNX_PreCalc_Figure_04_04_204.jpg)
![Graph of two functions, g(x) = ln(1/2)(x) in orange and f(x)=e^(x) in blue.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25201924/CNX_PreCalc_Figure_04_04_206.jpg)
![Graph of f(x)=log_2(x+2).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25201925/CNX_PreCalc_Figure_04_04_208.jpg)
![Graph of f(x)=ln(-x).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25201926/CNX_PreCalc_Figure_04_04_210.jpg)
![Graph of g(x)=log(6-3x)+1.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25201928/CNX_PreCalc_Figure_04_04_212.jpg)
![](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25201930/CNX_Precalc_Figure_04_04_219.jpg)
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