Question

Use properties of logarithms to rewrite each function, then graph. ƒ(x) = log3 [9 (x+2) ]

Accepted Answer

Recognize that the function is given as $$f(x) = \log_{3} \left[ 9 (x+2) \right]. $$The goal is to use logarithm properties to rewrite this expression in a simpler form. Recall the logarithm product property: $$\log_{a} (MN) = \log_{a} M + \log_{a} N. $$Apply this to separate the logarithm of the product inside the argument: $$f(x) = \log_{3} 9 + \log_{3} (x+2). $$Evaluate $$\log_{3} 9 by $$expressing 9 as a power of 3. Since $$9 = 3^2$, use the power property of logarithms: $$\log_{3} 9 = \log_{3} $$(3^2) = 2$. Substitute this value back into the expression to get f(x) = 2$$ + \log_{3} (x+2)$$, which is a simpler form of the original function. To graph f(x$$)$$, start with the graph of y$$ = \log_{3} (x+2)$$, which is a logarithmic function shifted 2 units to the left, then shift the entire graph up by 2 units to account for the constant term.

Use properties of logarithms to rewrite each function, then graph. ƒ(x) = log₃ [9 (x+2) ]

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