Test Your Knowledge: Solving Exponential Equations with Logarithms and Different Bases

📚 Quick Study Guide

• 🔢 Exponential Equation Form: An exponential equation takes the form $a^x = b$, where $a$ is the base, $x$ is the exponent, and $b$ is the result.
• 📝 Logarithm Definition: The logarithm (base $a$) of $b$ is the exponent to which $a$ must be raised to equal $b$. Mathematically, if $a^x = b$, then $\log_a(b) = x$.
• 🧮 Change of Base Formula: $\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$, where $c$ can be any convenient base, typically 10 or $e$ (natural logarithm). This is crucial when dealing with different bases.
• 💡 Solving Strategy: To solve $a^x = b$, take the logarithm of both sides (using the same base) or apply the change of base formula. Then, isolate $x$.
• ➕ Logarithm Properties: Useful properties include: $\log(mn) = \log(m) + \log(n)$, $\log(\frac{m}{n}) = \log(m) - \log(n)$, and $\log(m^p) = p \cdot \log(m)$.
• 🧪 Natural Logarithm: Denoted as $\ln(x)$, this is the logarithm with base $e$ (Euler's number, approximately 2.718). It is often used in calculus and other advanced math.

Practice Quiz

1. What is the value of $x$ in the equation $3^x = 27$?
   1. A) 2
   2. B) 3
   3. C) 4
   4. D) 9
2. Solve for $x$: $5^x = 125$
   1. A) 25
   2. B) 3
   3. C) 5
   4. D) 1
3. Solve for $x$: $2^x = 10$. (Round to two decimal places)
   1. A) 3.32
   2. B) 1.00
   3. C) 0.30
   4. D) 4.00
4. What is the value of $x$ in the equation $4^x = 16^{2}$?
   1. A) 2
   2. B) 4
   3. C) 6
   4. D) 8
5. Solve for $x$: $7^x = 49$
   1. A) 14
   2. B) 2
   3. C) 7
   4. D) 3
6. Solve for $x$: $6^x = 216$
   1. A) 2
   2. B) 36
   3. C) 3
   4. D) 4
7. What is the value of $x$ in the equation $9^x = 81$?
   1. A) 3
   2. B) 6
   3. C) 9
   4. D) 2