Algebra 2 exam prep: Converting log and exponent expressions

📚 Quick Study Guide

🔍 Logarithmic and exponential forms are inverses of each other. This means they "undo" each other. ➕ The logarithmic form is written as $\log_b(x) = y$, which is equivalent to the exponential form $b^y = x$. 🔄 $b$ is the base, $x$ is the argument (the value you're taking the logarithm of), and $y$ is the exponent. 💡 Remember: The base of the logarithm is the same as the base of the exponential expression. ✍️ To convert from logarithmic to exponential form, identify $b$, $x$, and $y$ in the logarithmic expression and plug them into the exponential form $b^y = x$. 📐 To convert from exponential to logarithmic form, identify $b$, $x$, and $y$ in the exponential expression and plug them into the logarithmic form $\log_b(x) = y$. ➗ The common logarithm has a base of 10. If the base is not written, it is assumed to be 10: $\log(x) = \log_{10}(x)$. 📈 The natural logarithm has a base of $e$ (Euler's number, approximately 2.71828): $\ln(x) = \log_e(x)$.

🧪 Practice Quiz

1. What is the exponential form of $\log_3(9) = 2$?
   1. $3^2 = 9$
   2. $2^3 = 9$
   3. $9^2 = 3$
   4. $9^3 = 2$
2. What is the logarithmic form of $5^3 = 125$?
   1. $\log_5(125) = 3$
   2. $\log_3(125) = 5$
   3. $\log_5(3) = 125$
   4. $\log_{125}(5) = 3$
3. Convert $\log_{2}(32) = 5$ to exponential form.
   1. $2^5 = 32$
   2. $5^2 = 32$
   3. $32^5 = 2$
   4. $2^{32} = 5$
4. Convert $7^2 = 49$ to logarithmic form.
   1. $\log_7(49) = 2$
   2. $\log_2(49) = 7$
   3. $\log_7(2) = 49$
   4. $\log_{49}(7) = 2$
5. What is the exponential form of $\ln(x) = 4$? (Remember that $\ln$ is the natural logarithm with base $e$)
   1. $e^4 = x$
   2. $4^e = x$
   3. $e^x = 4$
   4. $4^x = e$
6. What is the logarithmic form of $e^y = 10$? (Remember that $\ln$ is the natural logarithm with base $e$)
   1. $\ln(10) = y$
   2. $\ln(y) = 10$
   3. $\log_{10}(e) = y$
   4. $\log_y(e) = 10$
7. Solve for $x$: $\log_4(x) = 3$
   1. 64
   2. 7
   3. 12
   4. 81