AP Precalculus Flashcards: Exponential And Logarithmic Equations And Inequalities

What this deck covers

This deck focuses on Exponential And Logarithmic Equations And Inequalities, 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 the solution: $2^x = 16$.

Answer: $x = 4$. Since $16 = 2^4$, we have $x = 4$.

Flashcard 2: Solve the equation: $\text{ln}(x) = 2$.

Answer: $x = e^2$. Convert to exponential form: $e^2 = x$.

Flashcard 3: Identify the change of base formula for logarithms.

Answer: $\text{log}_b(x) = \frac{\text{log}_k(x)}{\text{log}_k(b)}$. Converts logarithm from base $b$ to any other base $k$.

Flashcard 4: Simplify: $e^{\text{ln}(x)}$.

Answer: $x$. Exponential and natural logarithm cancel as inverse functions.

Flashcard 5: What is the value of $\text{log}_b(b^0)$?

Answer:

1. Since $b^0 = 1$ and $\log_b(1) = 0$.

Flashcard 6: Solve for $x$: $e^x = 1$.

Answer: $x = 0$. Any number to the power of 0 equals 1.

Flashcard 7: What is the base change formula for $\text{log}_b(x)$?

Answer: $\text{log}_b(x) = \frac{\text{ln}(x)}{\text{ln}(b)}$. Uses natural logarithm to convert between bases.

Flashcard 8: Solve for $x$: $4^x = 64$.

Answer: $x = 3$. Since $64 = 4^3$, we have $x = 3$.

Flashcard 9: What does the equation $b^{\log_b(x)} = x$ represent?

Answer: Identity property of logarithms. Exponential and logarithm cancel each other as inverse functions.

Flashcard 10: Solve the equation: $x = \text{log}_2(32)$.

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

Flashcard 11: Solve for $x$: $10^x = 1000$.

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

Flashcard 12: Solve the inequality: $\text{ln}(x) < 0$.

Answer: $0 < x < 1$. Natural log is negative when input is between 0 and 1.

Flashcard 13: Simplify: $\text{log}_b(b^x)$.

Answer: $x$. Logarithm and exponential cancel each other as inverse functions.

Flashcard 14: Simplify: $\text{log}_b(b)$.

Answer:

1. Logarithm of the base itself always equals 1.

Flashcard 15: Find the solution: $\text{log}_3(x) = 0$.

Answer: $x = 1$. Convert to exponential: $3^0 = 1$, so $x = 1$.

Flashcard 16: What is the inverse of $y = 10^x$?

Answer: $y = \text{log}_{10}(x)$. Common logarithm is inverse of base-10 exponential function.

Flashcard 17: State the property: $\text{log}_b(x^r)$.

Answer: $r \text{log}_b(x)$. Power rule: exponent becomes coefficient in logarithm.

Flashcard 18: Define the natural logarithm function.

Answer: $\text{ln}(x) = \text{log}_e(x)$. Natural log uses base $e$ (Euler's number) as the base.

Flashcard 19: What is the domain of $\text{log}_b(x)$?

Answer: $x > 0$. Logarithm only defined for positive real numbers.

Flashcard 20: What is the inverse of $\text{ln}(x)$?

Answer: $e^x$. Natural logarithm and exponential function are inverses.

Flashcard 21: Solve for $x$: $5^x = 25$.

Answer: $x = 2$. Since $25 = 5^2$, we have $x = 2$.

Flashcard 22: What is the base of the common logarithm?

Answer:

1. Common logarithm uses base 10 by convention.

Flashcard 23: What is the value of $\text{ln}(1)$?

Answer:

1. Natural logarithm of 1 is always zero.

Flashcard 24: What is the range of the natural logarithm function?

Answer: All real numbers. Logarithm function outputs all real values for positive inputs.

Flashcard 25: What is the inverse of an exponential function $y = b^x$?

Answer: $x = \text{log}_b(y)$. Exponential and logarithmic functions are inverse operations.

Flashcard 26: What does the equation $b^{\text{log}_b(x)} = x$ represent?

Answer: Identity property of logarithms. Exponential and logarithm cancel each other as inverse functions.

Flashcard 27: Simplify: $\text{log}_a(1)$.

Answer:

1. Logarithm of 1 equals zero for any base.

Flashcard 28: State the base of the natural logarithm.

Answer: $e$. Natural logarithm uses Euler's number $e ≈ 2.718$ as base.

Flashcard 29: State the property: $\text{log}_b(xy)$.

Answer: $\text{log}_b(x) + \text{log}_b(y)$. Product rule: logarithm of product equals sum of logarithms.

Flashcard 30: What is the general form of an exponential function?

Answer: $f(x) = ab^x$ where $a \neq 0$, $b > 0$, $b \neq 1$. Standard form where $a$ is initial value and $b$ is growth/decay factor.

Flashcard 31: Solve the inequality: $3^x > 9$.

Answer: $x > 2$. Since $9 = 3^2$, we need $x > 2$.

Flashcard 32: Convert to exponential form: $\text{log}_b(x) = y$.

Answer: $b^y = x$. Definition of logarithm as the inverse of exponential function.

Flashcard 33: What is the range of $f(x) = b^x$ for $b > 1$?

Answer: $(0, \text{inf})$. Exponential functions with $b > 1$ produce all positive outputs.

Flashcard 34: What is the range of the function $f(x) = \text{log}_b(x)$?

Answer: All real numbers. Logarithmic functions output all real values.

Flashcard 35: State the property: $\text{log}_b(\frac{x}{y})$.

Answer: $\text{log}_b(x) - \text{log}_b(y)$. Quotient rule: logarithm of quotient equals difference of logarithms.

Flashcard 36: Identify the property: $b^{\text{log}_b(x)}$.

Answer: $x$. Identity property: base raised to its own logarithm.

Flashcard 37: What is $\text{log}_b(1)$ for any base $b$?

Answer:

1. Any base raised to power 0 equals 1.

Flashcard 38: Solve for $x$: $\text{log}_2(x) = 3$.

Answer: $x = 8$. Convert to exponential form: $2^3 = 8$, so $x = 8$.

Flashcard 39: State the formula for the natural exponential function.

Answer: $f(x) = e^x$. Uses Euler's number $e$ as the base for the exponential function.