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AP Precalculus Flashcards: Inverses Of Exponential Functions

Study Inverses Of Exponential Functions in AP Precalculus with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Precalculus

Inverses Of Exponential Functions

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QUESTION
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What operation is the inverse of exponentiation?

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ANSWER

The inverse operation is taking the logarithm. Logarithm is the inverse operation of exponentiation.

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What this deck covers

This deck focuses on Inverses Of Exponential Functions, 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.

All flashcards

Flashcard 1: What operation is the inverse of exponentiation?

Answer: The inverse operation is taking the logarithm. Logarithm is the inverse operation of exponentiation.

Flashcard 2: Identify the inverse of f(x)=axf(x) = a^x.

Answer: The inverse is f−1(x)=loga(x)f^{-1}(x) = \text{log}_a(x). Base-aa logarithm undoes base-aa exponentiation.

Flashcard 3: Identify the inverse of f(x)=9x−4f(x) = 9^{x-4}.

Answer: The inverse is f−1(x)=log9(x)+4f^{-1}(x) = \text{log}_9(x) + 4. Subtract 4 from exponent, so add 4 to inverse.

Flashcard 4: Find the inverse of f(x)=ex+2f(x) = e^{x+2}.

Answer: The inverse is f−1(x)=ln(x)−2f^{-1}(x) = \text{ln}(x) - 2. Addition in exponent becomes subtraction in inverse.

Flashcard 5: Find the inverse of f(x)=122x+1f(x) = 12^{2x+1}.

Answer: The inverse is f−1(x)=12log12(x)−12f^{-1}(x) = \frac{1}{2} \text{log}_{12}(x) - \frac{1}{2}. Factor out coefficient from exponent terms.

Flashcard 6: Identify the inverse of f(x)=b2x+1f(x) = b^{2x+1}.

Answer: The inverse is f−1(x)=12logb(x)−12f^{-1}(x) = \frac{1}{2} \text{log}_b(x) - \frac{1}{2}. Factor out coefficient and shift from exponent.

Flashcard 7: What is the inverse of f(x)=103xf(x) = 10^{3x}?

Answer: The inverse is f−1(x)=13log10(x)f^{-1}(x) = \frac{1}{3} \text{log}_{10}(x). Coefficient 3 in exponent becomes divisor 13\frac{1}{3}.

Flashcard 8: Which function is the inverse of f(x)=4x−3f(x) = 4^{x-3}?

Answer: The inverse is f−1(x)=log4(x)+3f^{-1}(x) = \text{log}_4(x) + 3. Subtract 3 from exponent, so add 3 to inverse.

Flashcard 9: Identify the steps to find inverse of f(x)=axf(x) = a^x.

Answer: Switch xx and yy, solve for yy. Result: y=loga(x)y = \text{log}_a(x). Standard method for finding inverse functions.

Flashcard 10: Identify the inverse of f(x)=2xf(x) = 2^x.

Answer: The inverse is f−1(x)=log2(x)f^{-1}(x) = \text{log}_2(x). Base-2 logarithm undoes base-2 exponentiation.

Flashcard 11: Identify the inverse of f(x)=15x−2f(x) = 15^{x-2}.

Answer: The inverse is f−1(x)=log15(x)+2f^{-1}(x) = \text{log}_{15}(x) + 2. Subtract 2 from exponent, so add 2 to inverse.

Flashcard 12: Find the inverse of f(x)=7x−1f(x) = 7^{x-1}.

Answer: The inverse is f−1(x)=log7(x)+1f^{-1}(x) = \text{log}_7(x) + 1. Subtract 1 from exponent, so add 1 to inverse.

Flashcard 13: What is the inverse of the natural exponential function?

Answer: The inverse is the natural logarithm function. Natural log ln⁡(x)\ln(x) undoes exe^x.

Flashcard 14: Find the inverse of f(x)=6x+1f(x) = 6^{x+1}.

Answer: The inverse is f−1(x)=log6(x)−1f^{-1}(x) = \text{log}_6(x) - 1. Addition in exponent becomes subtraction in inverse.

Flashcard 15: Which is the inverse of f(x)=9x−2f(x) = 9^{x-2}?

Answer: The inverse is f−1(x)=log9(x)+2f^{-1}(x) = \text{log}_9(x) + 2. Subtract 2 from exponent, so add 2 to inverse.

Flashcard 16: Convert y=2x−5y = 2^{x-5} to its inverse function.

Answer: The inverse is y=log2(x)+5y = \text{log}_2(x) + 5. Subtract 5 from exponent, so add 5 to inverse.

Flashcard 17: What is the inverse of the function f(x)=exf(x) = e^x?

Answer: The inverse is f−1(x)=ln(x)f^{-1}(x) = \text{ln}(x). Natural log undoes the natural exponential function.

Flashcard 18: Find the inverse of f(x)=5xf(x) = 5^x.

Answer: The inverse is f−1(x)=log5(x)f^{-1}(x) = \text{log}_5(x). Base-5 logarithm undoes base-5 exponentiation.

Flashcard 19: What is the inverse of f(x)=23xf(x) = 2^{3x}?

Answer: The inverse is f−1(x)=13log2(x)f^{-1}(x) = \frac{1}{3} \text{log}_2(x). Coefficient 3 in exponent becomes divisor 13\frac{1}{3}.

Flashcard 20: State the property used to find inverses of exponential functions.

Answer: Exponential and logarithmic functions are inverses. Each function undoes the other's operation.

Flashcard 21: Convert y=ex−1y = e^{x-1} to its inverse function.

Answer: The inverse is y=ln(x)+1y = \text{ln}(x) + 1. Subtract 1 from exponent, so add 1 to inverse.

Flashcard 22: What is the inverse function of f(x)=e2xf(x) = e^{2x}?

Answer: The inverse is f−1(x)=12ln(x)f^{-1}(x) = \frac{1}{2} \text{ln}(x). Coefficient in exponent becomes divisor in inverse.

Flashcard 23: What is the inverse of f(x)=82xf(x) = 8^{2x}?

Answer: The inverse is f−1(x)=12log8(x)f^{-1}(x) = \frac{1}{2} \text{log}_8(x). Coefficient 2 in exponent becomes divisor 12\frac{1}{2}.

Flashcard 24: Identify the inverse of f(x)=73x−1f(x) = 7^{3x-1}.

Answer: The inverse is f−1(x)=13(log7(x)+1)f^{-1}(x) = \frac{1}{3} (\text{log}_7(x) + 1). Factor out coefficient from both terms in exponent.

Flashcard 25: What is the inverse of f(x)=13xf(x) = 13^x?

Answer: The inverse is f−1(x)=log13(x)f^{-1}(x) = \text{log}_{13}(x). Base-13 logarithm undoes base-13 exponentiation.

Flashcard 26: Convert y=5x−4y = 5^{x-4} to its inverse function.

Answer: The inverse is y=log5(x)+4y = \text{log}_5(x) + 4. Subtract 4 from exponent, so add 4 to inverse.

Flashcard 27: Which function is the inverse of f(x)=3xf(x) = 3^x?

Answer: The inverse is f−1(x)=log3(x)f^{-1}(x) = \text{log}_3(x). Base-3 logarithm undoes base-3 exponentiation.

Flashcard 28: What is the inverse of f(x)=bxf(x) = b^x?

Answer: The inverse is f−1(x)=logb(x)f^{-1}(x) = \text{log}_b(x). General base-bb logarithm undoes base-bb exponentiation.

Flashcard 29: Determine the inverse of f(x)=2x+3f(x) = 2^{x+3}.

Answer: The inverse is f−1(x)=log2(x)−3f^{-1}(x) = \text{log}_2(x) - 3. Addition in exponent becomes subtraction in inverse.

Flashcard 30: Find the inverse of f(x)=3x+5f(x) = 3^{x+5}.

Answer: The inverse is f−1(x)=log3(x)−5f^{-1}(x) = \text{log}_3(x) - 5. Addition in exponent becomes subtraction in inverse.

Flashcard 31: Convert y=4xy = 4^x to its inverse function.

Answer: The inverse is x=4yx = 4^y or y=log4(x)y = \text{log}_4(x). Switch variables and solve for yy.

Flashcard 32: Find the inverse of f(x)=32xf(x) = 3^{2x}.

Answer: The inverse is f−1(x)=12log3(x)f^{-1}(x) = \frac{1}{2} \text{log}_3(x). Coefficient 2 in exponent becomes divisor 12\frac{1}{2}.

Flashcard 33: Find the inverse of f(x)=11x+2f(x) = 11^{x+2}.

Answer: The inverse is f−1(x)=log11(x)−2f^{-1}(x) = \text{log}_{11}(x) - 2. Addition in exponent becomes subtraction in inverse.

Flashcard 34: What is the inverse of f(x)=10xf(x) = 10^x?

Answer: The inverse is f−1(x)=log10(x)f^{-1}(x) = \text{log}_{10}(x). Common logarithm undoes base-10 exponentiation.