Logarithmic and exponential inverses Find the inverse of f(x)=b^(x) and then prove that they are inverses.

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Logarithmic and exponential inverses Find the inverse of  f(x)=b^(x) and then prove that they are inverses.

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Understand Inverse Function Concept: Understand the concept of an inverse function. An inverse function reverses the effect of the original function. If f(x) = y, then the inverse function f^(-1)(y) = x. For the function f(x) = b^x, we need to find a function that will give us the original x when we input y.Express Function in Terms of y: Express the function in terms of y. Let's write the original function with y instead of f(x): y = b^x.Solve for x in Terms of y: Solve for x in terms of y. To find the inverse, we need to solve for x: x = log_b(y). This means that the inverse function f^(-1)(y) is the logarithm base b of y.Write Inverse Function: Write the inverse function. The inverse function is f^(-1)(y) = log_b(y).Prove Functions are Inverses: Prove that f and f^(-1) are inverses. To prove that two functions are inverses, we need to show that f(f^(-1)(y)) = y and f^(-1)(f(x)) = x.Apply f to f^(-1): Apply f to f^(-1). Let's apply f to f^(-1)(y): f(f^(-1)(y)) = f(log_b(y)) = b^(log_b(y)).Simplify Expression: Simplify the expression. Using the property of logarithms that b^(log_b(y)) = y, we simplify the expression to get f(f^(-1)(y)) = y.Apply f^(-1) to f: Apply f^(-1) to f. Now let's apply f^(-1) to f(x): f^(-1)(f(x)) = f^(-1)(b^x) = log_b(b^x).Simplify Expression: Simplify the expression. Using the property of logarithms that log_b(b^x) = x, we simplify the expression to get f^(-1)(f(x)) = x.

Logarithmic and exponential inverses $\newline$ Find the inverse of $\newline$ $f(x)=b^{x}$ and then prove that they are inverses.

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Logarithmic and exponential inverses\newlineFind the inverse of \newlinef(x)=bxf(x)=b^{x}f(x)=bx and then prove that they are inverses.

Logarithmic and exponential inverses $\newline$ Find the inverse of $\newline$ $f(x)=b^{x}$ and then prove that they are inverses.