Problem 116 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The graph of \(f(x)=\log _{3} x\) contains the point (27,3).

Step-by-Step Solution

Answer

True. The statement is correct because 3 to the power of 3 equals 27.

Step 1: Understand the logarithmic function

The function in the exercise is a logarithm with base 3. In other words, the result of the function \(f(x) = \log_{3} x\) gives us the exponent to which we need to raise 3 to get x.

Step 2: Use the properties of logarithms

According to the properties of logarithms, if \(f(x) = \log_{b} x\), then \(b^{f(x)} = x\). Therefore, in order to check if the point (27,3) lies on the graph of the function \(f(x) = \log_{3} x\), we need to check whether raising 3 to the power of 3 equals 27.

Step 3: Calculate 3 to the power of 3

Calculate the value of 3 to the power of 3. This gives us 27.

Step 4: Check if the value of 3 to the power of 3 equals 27

Since 3 to the power of 3 equals 27, this confirms that the point (27,3) lies on the graph of \(f(x) = \log_{3} x\)