Problem 55 Find a formula for the inverse f... [FREE SOLUTION]

91影视

Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 3: Problem 55

Find a formula for the inverse function $f^{-1}$ of the indicated function $f$. $$ f(x)=4 \cdot 5^{x} $$

Short Answer

Expert verified

The inverse function of $f(x) = 4 \cdot 5^{x}$ is $f^{-1}(x) = \frac{\ln \left(\frac{x}{4}\right)}{\ln(5)}$.

Step by step solution

Replace f(x) with y

Let's replace $f(x)$ with $y$. We have: $$ y = 4 \cdot 5^{x} $$

Swap x and y

Now, let's switch the roles of $x$ and $y$, changing all $x$ to $y$ and all $y$ to $x$. We get: $$ x = 4 \cdot 5^{y} $$

Solve the equation for y

To solve the new equation for $y$, we will perform the following steps: 1. Divide by 4, 2. Take the natural logarithm, and 3. Divide by the natural logarithm of 5. First, divide both sides by 4: $$ \frac{x}{4} = 5^y $$ Next, take the natural logarithm of both sides: $$ \ln \left(\frac{x}{4}\right) = \ln (5^y) $$ Using the logarithm property $a \ln (b) = \ln (b^a)$, we can rewrite the right side as: $$ \ln \left(\frac{x}{4}\right) = y \ln(5) $$ Now, divide both sides by $\ln(5)$: $$ y = \frac{\ln \left(\frac{x}{4}\right)}{\ln(5)} $$ Finally, we express the inverse function as $f^{-1}(x)$: $$ f^{-1}(x) = \frac{\ln \left(\frac{x}{4}\right)}{\ln(5)} $$

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91影视

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=4 \cdot 5^{x} $$

Short Answer

Step by step solution

Replace f(x) with y

Swap x and y

Solve the equation for y

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