📚 Topic Summary
Solving exponential equations involves finding the value of a variable located in the exponent. The core strategy is to manipulate the equation so that both sides have the same base. Once the bases are equal, you can equate the exponents and solve for the variable. Sometimes, you'll need to use logarithms to isolate the variable if you can't easily get the same base on both sides.
🧠 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Exponential Function |
A. The inverse operation to exponentiation. |
| 2. Base |
B. The value that is raised to a power. |
| 3. Exponent |
C. A function of the form $f(x) = ab^x$, where $a$ is the initial value and $b$ is the growth/decay factor. |
| 4. Logarithm |
D. The power to which a base is raised. |
| 5. Growth Factor |
E. The factor by which a quantity increases over time in an exponential function; $b > 1$. |
Answers: 1-C, 2-B, 3-D, 4-A, 5-E
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct terms:
To solve the exponential equation $2^x = 8$, we need to express 8 as a power of ___(1)___. Since $8 = 2^3$, we can rewrite the equation as $2^x = 2^3$. Therefore, by equating the ___(2)___, we find that $x = ___(3)___. When the bases are not easily made the same, we can use ___(4)___ to solve. For example, to solve $5^x = 17$, we can take the logarithm of both sides, using either common or natural logarithms to isolate ___(5)___.
Answers: 1-2, 2-exponents, 3-3, 4-logarithms, 5-x
🤔 Part C: Critical Thinking
Explain in your own words why it is important to have the same base when solving exponential equations. What strategies can you use if the bases are different, and one can't be easily converted into the other?