Complete parts​ (a) and​ (b).

Question

a. Write the inverse of yequals11 Superscript x in logarithmic form.
b. Graph yequals11 Superscript x and its inverse and discuss the symmetry of their graphs.
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Part 1
a. The inverse of yequals11 Superscript x in logarithmic form is yequals
  
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Answer 1

To find the inverse of the function \( y = 11^x \) in logarithmic form, we first rewrite the equation in the standard form.

Part a:

1. Start with the equation: \[ y = 11^x \]

2. To find the inverse, swap \( x \) and \( y \): \[ x = 11^y \]

3. Now, solve for \( y \) in terms of \( x \):

   • Take the logarithm (base 11) of both sides: \[ \log_{11}(x) = y \]

Therefore, the inverse of \( y = 11^x \) in logarithmic form is: \[ y = \log_{11}(x) \]

Part b:

Graphical Representation:

• Graph of \( y = 11^x \):

  • This graph is an exponential curve that increases rapidly as \( x \) increases. It passes through the point \( (0, 1) \) since \( 11^0 = 1 \) and approaches the x-axis (but never touches it) as \( x \) approaches negative infinity.

• Graph of \( y = \log_{11}(x) \):

  • This graph is a logarithmic curve that increases slowly. It passes through the point \( (1, 0) \) since \( \log_{11}(1) = 0 \) and approaches the y-axis (but never touches it) as \( x \) approaches zero from the right.

Symmetry Discussion:

• The graphs of the functions \( y = 11^x \) and \( y = \log_{11}(x) \) are symmetric about the line \( y = x \).
• This symmetry arises because if you take any point \( (a, b) \) on the graph of \( y = 11^x \), you can find the point \( (b, a) \) on the graph of \( y = \log_{11}(x) \). Thus, they reflect across the line \( y = x \).

In summary:

• a. The inverse of \( y = 11^x \) in logarithmic form is \( y = \log_{11}(x) \).
• b. Both graphs exhibit symmetry about the line \( y = x \).