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]
