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Chapter 3: Problem 99

Determine whether the statement is true or false. Justify your answer. You can determine the graph of \(f(x)=\log _{6} x\) by graphing \(g(x)=6^{x}\) and reflecting it about the \(x\) -axis.

Short Answer

Expert verified
The statement is false. The graph of \(f(x) = \log_{6}x\) would be obtained by reflecting the graph of \(g(x)=6^{x}\) across the line \(y=x\), not the x-axis.

Step by step solution

01

Understand the relationship between logarithmic and exponential function

The relationship between \(f(x)=\log _{6} x\) and \(g(x)=6^{x}\) is not one of reflection across the x-axis, but one of reflection across the line \(y=x\). This difference happens because the logarithmic function is the inverse of the exponential function, not its reflection.
02

Graphical representation

If you were to graph \(g(x)=6^{x}\), you'd get a curve that starts very near to the x-axis when x is negative and then rises steeply as \(x\) increases. When \(f(x)=\log_{6}x\) is graphed, this line mirrors the curve of \(g(x)=6^{x}\) not about the x-axis, but across the line \(y=x\). As such, reflecting \(g(x)=6^{x}\) about the x-axis would yield a curve in the third and second quadrants, which does not match the graph of \(f(x)=\log_{6}x\).

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