AP Precalculus Flashcards: Exponential Function Manipulation

What this deck covers

This deck focuses on Exponential Function Manipulation, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the inverse of the exponential function $f(x) = a^x$?

Answer: $f^{-1}(x) = \text{log}_a(x)$. Exponential and logarithmic functions are inverse pairs.

Flashcard 2: How do you express $e^{\text{ln}(x)}$ in terms of $x$?

Answer: $x$. The exponential and natural logarithm functions are inverses.

Flashcard 3: What is the expression $\text{e}^{x+y}$ equal to in exponential terms?

Answer: $\text{e}^x \times \text{e}^y$. Using the product rule for exponentials with the same base.

Flashcard 4: What is the general form of an exponential function?

Answer: $f(x) = a \times b^x$. Standard exponential form where $a$ is the initial value and $b$ is the base.

Flashcard 5: What is the y-intercept of $f(x) = a \times b^x$?

Answer: $a$. When $x = 0$, $f(0) = a \times b^0 = a \times 1 = a$.

Flashcard 6: What is the value of $f(x) = 4 \times (0.5)^x$ at $x = 2$?

Answer: $1$. Calculate: $4 \times (0.5)^2 = 4 \times 0.25 = 1$.

Flashcard 7: Evaluate $3^x = 243$ for $x$.

Answer: $x = 5$. Since $3^5 = 243$, we have $x = 5$.

Flashcard 8: What property of exponents is used in $a^x \times a^y = a^{x+y}$?

Answer: Product of powers. When multiplying powers with the same base, add the exponents.

Flashcard 9: Convert the expression $x = \ln(y)$ to exponential form.

Answer: $e^x = y$. Converting from logarithmic to exponential form.

Flashcard 10: What is $\text{ln}(\text{e}^x)$?

Answer: $x$. Natural logarithm and $e$ are inverse functions.

Flashcard 11: What is the range of the function $f(x) = 3^x$?

Answer: $(0, \text{inf})$. Exponential functions with positive bases have all positive outputs.

Flashcard 12: What is the expression for $f(x) = 2 \times b^x$ when $f(x) = 16$?

Answer: $2 \times b^x = 16$. Setting the function equal to the given value creates an equation.

Flashcard 13: What is the base $b$ in $f(x) = b^x$ for $b > 0$ and $b \neq 1$?

Answer: $b$ is a positive constant not equal to 1. Base must be positive and not equal to 1 for exponential functions.

Flashcard 14: What is the expression for $e^{2\text{ln}(x)}$?

Answer: $x^2$. Using the property $e^{\ln(x)} = x$ and power rule.

Flashcard 15: What is $\text{log}_b(b^x)$ equal to?

Answer: $x$. The logarithm and exponential with the same base cancel out.

Flashcard 16: What is the exponential form of $x = \text{log}_b(y)$?

Answer: $b^x = y$. Converting from logarithmic to exponential form.

Flashcard 17: What does the expression $b^{x-y}$ simplify to?

Answer: $\frac{b^x}{b^y}$. Subtracting exponents equals division of the same base powers.

Flashcard 18: Which rule is used to solve $b^{x+y} = b^x \times b^y$?

Answer: Exponential rule for addition. This is the fundamental property for combining exponential expressions.

Flashcard 19: What is the derivative of $f(x) = e^x$?

Answer: $f'(x) = e^x$. The natural exponential function is its own derivative.

Flashcard 20: What is the simplified form of $b^0$?

Answer: $1$. Any non-zero number raised to the power 0 equals 1.

Flashcard 21: How do you express $\text{log}_a(a^x)$?

Answer: $x$. The logarithm and exponential with the same base cancel out.

Flashcard 22: How do you solve $e^x = 5$?

Answer: $x = \text{ln}(5)$. Taking the natural logarithm of both sides isolates $x$.

Flashcard 23: How do you solve $3^x = 27$?

Answer: $x = 3$. Since $3^3 = 27$, we have $x = 3$.

Flashcard 24: Rewrite the expression $b^{-x}$ in terms of fractions.

Answer: $\frac{1}{b^x}$. Negative exponents represent reciprocals of positive exponents.

Flashcard 25: Convert $8 = 2^x$ into a logarithmic equation.

Answer: $x = \text{log}_2(8)$. Converting from exponential to logarithmic form.

Flashcard 26: What is $\text{log}_a(1)$ equal to?

Answer: $0$. Any number raised to the power 0 equals 1, so $\log_a(1) = 0$.

Flashcard 27: Evaluate $2^x = 32$ for $x$.

Answer: $x = 5$. Since $2^5 = 32$, we have $x = 5$.

Flashcard 28: What is the simplified form of $(b^x)^0$?

Answer: $1$. Any expression raised to the power 0 equals 1.

Flashcard 29: What is the value of $f(x) = 5 \times 2^x$ when $x = 3$?

Answer: $40$. Substitute $x = 3$: $f(3) = 5 \times 2^3 = 5 \times 8 = 40$.

Flashcard 30: Find the value of $x$ in $10^x = 1000$.

Answer: $x = 3$. Since $10^3 = 1000$, we have $x = 3$.

Flashcard 31: What is $f(x) = 2^x$ expressed as a logarithmic function?

Answer: $x = \text{log}_2(f(x))$. Taking the logarithm base 2 of both sides isolates $x$.

Flashcard 32: What is the value of $x$ if $2^x = 1$?

Answer: $x = 0$. Any positive number raised to power 0 equals 1.

Flashcard 33: How do you rewrite $4^x = 16$ using logarithms?

Answer: $x = \text{log}_4(16)$. Converting exponential to logarithmic form by taking log of both sides.

Flashcard 34: What is the simplified form of $\text{log}_b(b)$?

Answer: $1$. The logarithm of a base to itself equals 1.

Flashcard 35: Convert the expression $x = \text{ln}(y)$ to exponential form.

Answer: $\text{e}^x = y$. Converting from logarithmic to exponential form.

Flashcard 36: What is $b^1$ equal to?

Answer: $b$. Any number raised to the power 1 equals itself.

Flashcard 37: What is the domain of $f(x) = 2^x$?

Answer: $(-\text{inf}, \text{inf})$. Exponential functions accept all real number inputs.

Flashcard 38: Express $b^{x+y}$ in its expanded form.

Answer: $b^x \times b^y$. The sum rule for exponents breaks into separate factors.

Flashcard 39: Identify the base in the function $f(x) = 3 \times 2^x$.

Answer:

1. The base is the number being raised to the power $x$.

Flashcard 40: Simplify the expression $(b^x)^y$.

Answer: $b^{xy}$. Power of a power rule: multiply the exponents.

Flashcard 41: What is the inverse of $f(x) = \text{e}^x$?

Answer: $f^{-1}(x) = \text{ln}(x)$. The natural exponential and logarithm are inverse functions.