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Find the inverse of the function.

[tex] y = \log_5(2x) [/tex]

The inverse is [tex] y = \square [/tex] (Simplify your answer.)


Sagot :

To find the inverse of the function [tex]\( y = \log_5(2x) \)[/tex], follow these detailed steps:

1. Start with the given function:
[tex]\[ y = \log_5(2x) \][/tex]

2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse:
[tex]\[ x = \log_5(2y) \][/tex]

3. Rewrite the equation in its exponential form to solve for [tex]\( y \)[/tex]:
Recall that if [tex]\( a = \log_b(c) \)[/tex], then [tex]\( b^a = c \)[/tex]. Using this property, convert the logarithmic equation into an exponential equation:
[tex]\[ 5^x = 2y \][/tex]

4. Solve for [tex]\( y \)[/tex]:
To isolate [tex]\( y \)[/tex], divide both sides of the equation by 2:
[tex]\[ y = \frac{5^x}{2} \][/tex]

Thus, the inverse of the function [tex]\( y = \log_5(2x) \)[/tex] is:
[tex]\[ y = \frac{5^x}{2} \][/tex]
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