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Logarithmic Functions

Logarithmic scales are used when the range of possible values is very wide, such as the intensity of an earthquake or the acidity of a liquid, in order to avoid the use of large numbers.

Graphs

Let \(f\) be the translation \(f: (x,y) \to (x+m, y+n)\), where \(m\) and \(n\) are constants. If \(f\) transforms the curve \(y=\log_2 3x\) to \(y=\log_2 (12x-108)\), what is \(m+n\)?

Which of the following could be the graph of \( \log_a {x} \: \) if \( 0<a<1 \:? \)

Which of the following could be the equation of the above graph?

(A) \(~ y = \log{|x|} \)
(B) \(~ y = \left|\log{|x|}\right| \)
(C) \(~ |y| = \log{x} \)
(D) \(~ y = |\log{x}| \)

For \(x\) and \(y\) satisfying \(x+y=8\), let \(m\) be the maximum value of \[\log_{4}x+\log_{4}y.\] If \(a\) and \(b\) are the values of \(x\) and \(y\), respectively, for which the maximum value is attained, what is \(m+2a+3b\)?

Which of the following could be the graph of \( \log_a {x} \: \) if \( 0<a<1 \:? \)

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