To find the inverse of \( y = 3^x \) in logarithmic form, we first need to express \( x \) in terms of \( y \).
Starting with the equation:
\[ y = 3^x \]
We can rewrite this in logarithmic form:
\[ x = \log_3(y) \]
Since we are looking for the inverse function (which we can denote as \( y^{-1} \) or simply \( f^{-1}(y) \)), we can express it as:
\[ y = \log_3(x) \]
So the inverse of \( y = 3^x \) in logarithmic form is:
\[ y = \log_3(x) \]
Part b: Graphing \( y = 3^x \) and its inverse
Now let's discuss the graphs of \( y = 3^x \) and \( y = \log_3(x) \).
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Graph of \( y = 3^x \):
- This is an exponential growth function. It passes through the point (0, 1) because \( 3^0 = 1 \).
- As \( x \) increases, \( y \) increases rapidly, and as \( x \) decreases, \( y \) approaches 0 but never reaches it (asymptote at \( y = 0 \)).
- The graph is always above the x-axis.
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Graph of \( y = \log_3(x) \):
- This is the inverse function of \( y = 3^x \). It passes through the point (1, 0) since \( \log_3(1) = 0 \).
- As \( x \) increases, \( y \) increases slowly, and it approaches \( \infty \) as \( x \) approaches \( \infty \).
- As \( x \) approaches 0 from the right, \( y \) decreases without bound (it approaches \( -\infty \)).
- The graph is only defined for \( x > 0 \).
Symmetry: The graphs of \( y = 3^x \) and \( y = \log_3(x) \) are reflections of each other across the line \( y = x \). This symmetry indicates that if you were to fold the graph along the line \( y = x \), the two graphs would match up exactly.
Thus, the points on the graph of \( y = 3^x \) correspond to points on the graph of \( y = \log_3(x) \) with their \( x \) and \( y \) values swapped (for example, the point (0, 1) on \( y = 3^x \) corresponds to (1, 0) on \( y = \log_3(x) \)).
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