AP Precalculus Flashcards: Logarithmic Functions

What this deck covers

This deck focuses on Logarithmic 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.

Flashcard 1: Simplify $\text{log}_b(1)$.

Answer:

1. Any base raised to the power 0 equals 1.

Flashcard 2: What is $\text{log}_{10}(100)$?

Answer:

1. Since $10^2 = 100$, the answer is 2.

Flashcard 3: Find $x$ if $\text{log}_3(x) = 4$.

Answer:

1. Convert to exponential: $3^4 = 81$.

Flashcard 4: Determine the range of $f(x) = \text{log}_b(x)$.

Answer: All real numbers. Logarithms can output any real value as input approaches 0 or infinity.

Flashcard 5: What does $\text{log}_b(0)$ evaluate to?

Answer: Undefined. Logarithm of zero does not exist in real numbers.

Flashcard 6: What is the definition of a logarithm?

Answer: If $b^x = y$, then $\text{log}_b(y) = x$. This defines the inverse relationship between exponentials and logarithms.

Flashcard 7: What is $\text{ln}(1)$?

Answer:

1. Natural log of 1 equals 0 since $e^0 = 1$.

Flashcard 8: What is the inverse function of $y = \text{log}_b(x)$?

Answer: $x = b^y$. Inverse functions undo each other's operations.

Flashcard 9: If $f(x) = \text{log}_b(x)$, find $f(b^4)$.

Answer:

1. Logarithm and exponential with same base cancel out.

Flashcard 10: State the logarithm power rule.

Answer: $\text{log}_b(x^n) = n \times \text{log}_b(x)$. Exponents become multipliers when using logarithms.

Flashcard 11: What is the change of base formula?

Answer: $\text{log}_b(x) = \frac{\text{log}_k(x)}{\text{log}_k(b)}$. Converts between different logarithm bases using any base $k$.

Flashcard 12: What is the value of $\text{log}_a(a)$?

Answer:

1. Any base raised to power 1 equals itself.

Flashcard 13: What is $\text{log}_3(1)$?

Answer:

1. Any base raised to the power 0 equals 1.

Flashcard 14: What is the simplified form of $\text{log}_b(b^x)$?

Answer: $x$. Logarithm and exponential with same base cancel out.

Flashcard 15: Find the value of $\text{log}_7(49)$.

Answer:

1. Since $7^2 = 49$, the answer is 2.

Flashcard 16: Express $\text{log}_b(x^3)$ using the power rule.

Answer: $3 \times \text{log}_b(x)$. Power rule moves the exponent as a coefficient.

Flashcard 17: What does $\text{ln}(e)$ equal?

Answer:

1. Natural log of its own base equals 1.

Flashcard 18: Convert $\text{log}_b(\frac{x}{y})$ using the quotient rule.

Answer: $\text{log}_b(x) - \text{log}_b(y)$. Quotient rule expands division into subtraction of logs.

Flashcard 19: What is the base of the natural logarithm?

Answer: The base of the natural logarithm is $e$. $e ≈ 2.718$ is Euler's number, the natural base.

Flashcard 20: Express $10^x = 1000$ in logarithmic form.

Answer: $\text{log}_{10}(1000) = x$. Direct conversion from exponential to logarithmic form.

Flashcard 21: Evaluate $\text{log}_5(25)$.

Answer:

1. Since $5^2 = 25$, the answer is 2.

Flashcard 22: Identify the domain of $f(x) = \text{log}_b(x)$.

Answer: $x > 0$. Logarithms are only defined for positive arguments.

Flashcard 23: Find $\text{log}_b(b^x)$.

Answer: x. Logarithm and exponential with same base cancel out.

Flashcard 24: Simplify $\text{log}_a(a^5)$.

Answer:

1. Logarithm and exponential with same base cancel out.

Flashcard 25: Express the exponential form of $\text{log}_5(125) = 3$.

Answer: $5^3 = 125$. Direct conversion from logarithmic to exponential form.

Flashcard 26: Convert $\text{ln}(e^3)$ to a simpler form.

Answer:

1. Natural log and $e$ with same exponent cancel out.

Flashcard 27: What is the value of $\text{log}_b(b)$?

Answer:

1. Any base raised to the power 1 equals itself.

Flashcard 28: Convert the exponential equation $b^x = y$ to logarithmic form.

Answer: $\text{log}_b(y) = x$. Direct conversion using the definition of logarithms.

Flashcard 29: Simplify $\text{log}_b(\frac{b^2}{b^3})$.

Answer: -1. Quotient simplifies to $b^{-1}$, so $\text{log}_b(b^{-1}) = -1$.

Flashcard 30: Express $\text{log}_b(\frac{1}{x})$ using logarithm rules.

Answer: $-\text{log}_b(x)$. Reciprocal creates negative exponent, so negative log.

Flashcard 31: Simplify $\log_a(a^5)$.

Answer:

1. Logarithm and exponential with same base cancel out.

Flashcard 32: Express $e^{\text{ln}(x)}$ in simpler form.

Answer: x. Exponential and natural log are inverse functions.

Flashcard 33: What is the logarithm quotient rule?

Answer: $\text{log}_b(\frac{x}{y}) = \text{log}_b(x) - \text{log}_b(y)$. Quotients inside logs become differences of logs.

Flashcard 34: Evaluate $\text{log}_{10}(0.1)$.

Answer: -1. Since $10^{-1} = 0.1$, the answer is -1.

Flashcard 35: What is the base of the common logarithm?

Answer:

1. Common log uses base 10 by convention.

Flashcard 36: Express $\text{log}_b(a^n)$ using the power rule.

Answer: $n \times \text{log}_b(a)$. Power rule moves exponent $n$ as coefficient.

Flashcard 37: What is $\text{log}_b(b^0)$?

Answer:

1. Since $b^0 = 1$ and $\text{log}_b(1) = 0$.

Flashcard 38: Solve for $x$: $\text{log}_4(64) = x$.

Answer:

1. Since $4^3 = 64$, the answer is 3.

Flashcard 39: What is the logarithm product rule?

Answer: $\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)$. Products inside logs become sums of logs.

Flashcard 40: Evaluate $\text{log}_2(8)$.

Answer:

1. Since $2^3 = 8$, the answer is 3.