Mastering Exponential Function Inverses: Top Tips and Examples

📚 Quick Study Guide

• 🔢 An exponential function has the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.
• 🔄 The inverse of an exponential function is a logarithmic function.
• 📝 If $f(x) = a^x$, then $f^{-1}(x) = \log_a(x)$.
• 📈 To find the inverse, swap $x$ and $y$ and solve for $y$. For example, if $y = 2^x$, then $x = 2^y$, and $y = \log_2(x)$.
• 💡 Key properties of logarithms include: $\log_a(1) = 0$, $\log_a(a) = 1$, and $\log_a(a^x) = x$.
• ➗ Remember the change of base formula: $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$. This is useful for calculations!
• 🧠 Exponential functions are one-to-one, ensuring their inverses are also functions.

Practice Quiz

1. What is the inverse of the function $f(x) = 3^x$?
   1. $f^{-1}(x) = x^3$
   2. $f^{-1}(x) = \log_3(x)$
   3. $f^{-1}(x) = -3^x$
   4. $f^{-1}(x) = \frac{1}{3^x}$
2. Find the inverse of $f(x) = e^x$.
   1. $f^{-1}(x) = \frac{1}{e^x}$
   2. $f^{-1}(x) = \ln(x)$
   3. $f^{-1}(x) = -e^x$
   4. $f^{-1}(x) = 10^x$
3. What is the value of $\log_5(25)$?
   1. 5
   2. 2
   3. 125
   4. 0
4. If $f(x) = 4^x$, what is $f^{-1}(16)$?
   1. 2
   2. 4
   3. \frac{1}{4}
   4. 64
5. Solve for $x$: $\log_2(x) = 3$.
   1. 6
   2. 8
   3. 9
   4. 5
6. What is the inverse of $f(x) = 10^x$?
   1. $f^{-1}(x) = \log_{10}(x)$
   2. $f^{-1}(x) = \ln(x)$
   3. $f^{-1}(x) = \frac{1}{10^x}$
   4. $f^{-1}(x) = x^{10}$
7. Simplify: $\log_a(a^7)$.
   1. $a$
   2. 7
   3. $a^7$
   4. 1