AP Precalculus Flashcards: Logarithmic Function Manipulation

What this deck covers

This deck focuses on Logarithmic 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: Find $\text{log}_3(27)$.

Answer:

1. Since $3^3 = 27$.

Flashcard 2: Solve for $x$: $\text{log}_4(x) = 2$.

Answer: $x = 16$. Convert to exponential: $4^2 = 16$.

Flashcard 3: State the base of the common logarithm.

Answer:

1. Base 10 logarithms are called common logarithms.

Flashcard 4: What is the Product Rule for logarithms?

Answer: $\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)$. Addition corresponds to multiplication inside logs.

Flashcard 5: What is the definition of a logarithm?

Answer: If $b^y = x$, then $\text{log}_b(x) = y$. A logarithm asks: what power of $b$ gives $x$?

Flashcard 6: Express $\text{log}_4(16)$ as a simple integer.

Answer:

1. Since $4^2 = 16$.

Flashcard 7: What is the value of $\text{log}_2(32)$?

Answer:

1. Since $2^5 = 32$.

Flashcard 8: Evaluate $\text{log}_2(1)$. What is the result?

Answer:

1. Any base raised to power 0 equals 1.

Flashcard 9: Simplify $\text{log}_5(1)$. What is the value?

Answer:

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

Flashcard 10: What is the logarithmic form of the equation $10^3 = 1000$?

Answer: $\text{log}_{10}(1000) = 3$. Convert by asking: what power of 10 gives 1000?

Flashcard 11: Simplify $\text{log}_b(1)$. What is the value?

Answer:

1. Any base raised to power 0 equals 1.

Flashcard 12: What is the value of $\text{log}_5(25)$?

Answer:

1. Since $5^2 = 25$.

Flashcard 13: Solve for $x$: $\text{log}_5(x) = 3$.

Answer: $x = 125$. Convert to exponential: $5^3 = 125$.

Flashcard 14: Evaluate $\log_{10}(100)$.

Answer:

1. Since $10^2 = 100$.

Flashcard 15: What is the Quotient Rule for logarithms?

Answer: $\text{log}_b\bigg(\frac{x}{y}\bigg) = \text{log}_b(x) - \text{log}_b(y)$. Subtraction corresponds to division inside logs.

Flashcard 16: Express $2 \times \text{log}_b(3)$ using a logarithm property.

Answer: $\text{log}_b(3^2)$. Use the power rule in reverse.

Flashcard 17: What is the inverse function of $y = \text{log}_b(x)$?

Answer: $y = b^x$. Logarithm and exponential functions are inverses.

Flashcard 18: What is the value of $\text{log}_2(8)$?

Answer:

1. Since $2^3 = 8$.

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

Answer: $e$. Natural logarithms use Euler's number as base.

Flashcard 20: Convert $\text{log}_b(x) - \text{log}_b(y)$ using a property.

Answer: $\text{log}_b\bigg(\frac{x}{y}\bigg)$. Use the quotient rule in reverse.

Flashcard 21: State the Power Rule for logarithms.

Answer: $\text{log}_b(x^n) = n \times \text{log}_b(x)$. The exponent comes down as a coefficient.

Flashcard 22: Simplify $\text{log}_3(27)$ to an integer.

Answer:

1. Since $3^3 = 27$.

Flashcard 23: What is $\text{log}_b(0)$?

Answer: Undefined. Cannot take logarithm of zero or negative numbers.

Flashcard 24: If $\text{log}_b(x) = y$, express $x$ in terms of $b$ and $y$.

Answer: $x = b^y$. Definition of logarithm in exponential form.

Flashcard 25: If $\text{log}_b(8) = 3$, find $b$.

Answer: $b = 2$. Since $2^3 = 8$.

Flashcard 26: What is the natural logarithm of $e$?

Answer:

1. By definition, $\ln(e) = 1$.

Flashcard 27: Evaluate $\text{log}_{10}(100)$.

Answer:

1. Since $10^2 = 100$.

Flashcard 28: Convert $10^x = 100$ to logarithmic form.

Answer: $x = \text{log}_{10}(100)$. Ask: what power of 10 gives 100?

Flashcard 29: Convert $\text{log}_a(x) + \text{log}_a(y)$ using a logarithm property.

Answer: $\text{log}_a(xy)$. Use the product rule in reverse.

Flashcard 30: What is the Change of Base Formula for logarithms?

Answer: $\text{log}_b(a) = \frac{\text{log}_c(a)}{\text{log}_c(b)}$. Allows conversion between different logarithm bases.

Flashcard 31: Convert $\text{log}_2(32)$ to exponential form.

Answer: $2^5 = 32$. Convert by writing as $b^y = x$ form.

Flashcard 32: Simplify $\text{log}_b(b^7)$. What is the value?

Answer:

1. The inverse property: $\text{log}_b(b^x) = x$.

Flashcard 33: Evaluate $\text{log}_{10}(1000)$.

Answer:

1. Since $10^3 = 1000$.

Flashcard 34: Convert $b^4 = 81$ to logarithmic form.

Answer: $\text{log}_b(81) = 4$. Ask: what power of $b$ gives 81?

Flashcard 35: What is the value of $\text{log}_{10}(1)$?

Answer:

1. Any base raised to power 0 equals 1.

Flashcard 36: What is $\text{log}_b(b)$ equal to?

Answer:

1. Any base to the first power equals itself.

Flashcard 37: If $\text{log}_3(x) = 4$, what is $x$?

Answer: $x = 81$. Convert to exponential form: $3^4 = 81$.

Flashcard 38: If $\text{log}_b(9) = 2$, what is $b$?

Answer: $b = 3$. Since $3^2 = 9$.