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This is one of our many quizzes on Graphing compound inequalities. Click on this to access more such competitive quizzes on Differential Calculus.

Graphing logarithmic functions Quiz 2

mathematics

 State the domain, range and the vertical asymptote of the function  $y = {\log _4}(2x - 8) + 2$ for which the graph is <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/3c4c6d5a-be4f-48d3-9d15-fe02bfb561012743585808180859179.png"/>

 So for finding the domain and range we will write: $ \\  2x - 8 > 0   \\  2x > 8   \\  x > 4   \\  $ So the domain will be $\left( {4,\infty } \right)$and the range will be $( - \infty ,\infty )$ $ \\ $ And as the $x > 4$ so the vertical asymptote will be $x = 4$. 

 Solve the logarithmic function in terms of $b$. The function is : ${\log _{\sqrt b }}x$. Also graph the function when $b = 2$.

 Use the logarithmic tricks ${\log _b}a = \dfrac{{\log a}}{{\log b}}$ and $\log {a^b} = b\log a$ to substitute under-root. 

 Applying the property tricks $ \\ {\log _b}a = \dfrac{{\log a}}{{\log b}} \\ \log {a^b} = b\log a \\ {\log _{\sqrt b }}x = \dfrac{{\log x}}{{\log \sqrt b }} \\ \Rightarrow \dfrac{{\log x}}{{\dfrac{1}{2}\log b}} \\ \Rightarrow 2\dfrac{{\log x}}{{\log b}} \\ \Rightarrow 2{\log _b}x \\ $ The graph for the function is as follows: $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/b930c57d-5926-43fa-9094-6387ce7ea4c64026683618942075137.png"/>

  State the vertical asymptote of the function $y = {\log _{\dfrac{1}{4}}}(8x - 6) + 4$

 To find the vertical asymptote, we will find the domain and range first. So, $ \\ \Rightarrow 8x - 6 > 0   \\   \Rightarrow 8x > 6   \\   \Rightarrow x > \dfrac{6}{8}   \\   \Rightarrow x > \dfrac{3}{4}   \\  $ So the domain is $\left( {\dfrac{3}{4},\infty } \right)$ and range is $( - \infty ,\infty )$, so the vertical asymptote will be $x = \dfrac{3}{4}$

 Solve the function in terms of $\sqrt x $. The function given is ${\log _{{b^4}}}{x^2}$. Graph the function when $b = 5$.

 Use the logarithmic trick $\log {a^b} = b\log a$.

 Applying the formula $\log {a^b} = b\log a$ $ \\ {\log _{{b^4}}}{x^2} = \dfrac{{\log {x^2}}}{{\log {b^4}}} \\ \Rightarrow \dfrac{{2\log x}}{{4\log b}} \\ \Rightarrow \dfrac{{1\log x}}{{2\log b}} \\ \Rightarrow {\log _b}\sqrt x \\ $ The graph for the function is as follows: $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/1ff6070b-c94b-45e5-a877-a2f6913bf9a26636745428295517189.png"/>

 Graph the function $y = {\log _6}(x - 1) - 1$. Check how many y-intercepts are there in the function. 

 The graph for the function is as follows: $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/3fe9e1c4-6874-4194-a68c-9fba84b58eb63398588190158499454.png"/>  $ \\ $ To find the y-intercept we need to put $x=0$ so, $ \\  y = {\log _6}(0 - 1) - 1   \\  y = {\log _6}( - 1) - 1   \\  $ So the function has no y-intercept. 