AP Precalculus Flashcards: Logarithmic Expressions

What this deck covers

This deck focuses on Logarithmic Expressions, 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 quotient property of logarithms for $\log_b\left(\frac{M}{N}\right)$?

Answer: $\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$. The log of a quotient equals the difference of the logs.

Flashcard 2: Identify the condensed form of $2\log_b(x)-\log_b(y)+3\log_b(z)$.

Answer: $\log_b\!\left(\frac{x^2z^3}{y}\right)$. Combine using power rule, then product/quotient rules.

Flashcard 3: Solve for $x$: $\log_7(x)=\frac{2}{3}$.

Answer: $x=7^{\frac{2}{3}}$. Convert to exponential form: $x=7^{2/3}$.

Flashcard 4: Identify the simplified value: $\log_3\left(\frac{1}{27}\right)$ equals what?

Answer: $-3$. $\frac{1}{27} = 3^{-3}$, so $\log_3(\frac{1}{27}) = -3$.

Flashcard 5: What is the one-to-one property of logarithms?

Answer: $\log_b(M)=\log_b(N)\iff M=N$. Equal logs with same base imply equal arguments.

Flashcard 6: What is the domain of $g(x)=\log_3(x^2-9)$?

Answer: $x<-3$ or $x>3$. Set $x^2-9>0$; factor as $(x-3)(x+3)>0$.

Flashcard 7: What is the natural logarithm notation: $\ln(x)$ means $\log_b(x)$ with what base?

Answer: $\ln(x)=\log_e(x)$. Natural log has base $e$ (Euler's number).

Flashcard 8: What is the domain condition for $log_b(a)$ in real numbers (for $a$ and $b$)?

Answer: $a>0,\ b>0,\ b\ne 1$. Base and argument must be positive; base cannot equal 1.

Flashcard 9: What is the common logarithm notation: $\log(x)$ means $\log_b(x)$ with what base?

Answer: $\log(x)=\log_{10}(x)$. Common log has base 10 by convention.

Flashcard 10: What is $\log_b(b)$ for any valid base $b$?

Answer: $\log_b(b)=1$. Any base raised to power 1 equals itself.

Flashcard 11: What is the exact value of $\log_2(32)$?

Answer: $5$. Since $2^5=32$, the answer is 5.

Flashcard 12: Identify the expanded form of $\log_b\!\left(\frac{x^3y}{\sqrt{z}}\right)$.

Answer: $3\log_b(x)+\log_b(y)-\frac{1}{2}\log_b(z)$. Apply product, quotient, and power rules; $\sqrt{z}=z^{1/2}$.

Flashcard 13: What is the power property of logarithms?

Answer: $\log_b(M^k)=k\log_b(M)$. Exponents come out front as coefficients.

Flashcard 14: What is the simplified value: $\ln(e^{-4})$ equals what?

Answer: $-4$. $\ln$ and $e$ are inverse functions.

Flashcard 15: What is the simplified value: $10^{\log(3)}$ equals what?

Answer: $3$. $10$ and $\log$ are inverses, so they cancel.

Flashcard 16: What is the inverse property: $\log_b(b^x)$ equals what (for real $x$)?

Answer: $\log_b(b^x)=x$. Log and exponential functions are inverses.

Flashcard 17: What is the exact value of $\log_5\!\left(\frac{1}{25}\right)$?

Answer: $-2$. Since $5^{-2}=\frac{1}{25}$, the answer is -2.

Flashcard 18: What is the value of $\log_b(1)$ for any valid base $b$?

Answer: $\log_b(1)=0$. Because $b^0=1$ for any valid base $b$.

Flashcard 19: What is the power property of logarithms for $\log_b(M^p)$?

Answer: $\log_b(M^p)=p\log_b(M)$. Exponents come out front as coefficients.

Flashcard 20: What is the expanded form of $\log_5\left(\frac{25x^3}{y}\right)$?

Answer: $2+3\log_5(x)-\log_5(y)$. Apply quotient, power, and $\log_5(25) = 2$.

Flashcard 21: What is $\log_b(1)$ for any valid base $b$?

Answer: $\log_b(1)=0$. Any base raised to power 0 equals 1.

Flashcard 22: What is the condensed form of $\log_2(x)+\log_2(y)-\log_2(4)$?

Answer: $\log_2\left(\frac{xy}{4}\right)$. Product property combines logs; quotient property for subtraction.

Flashcard 23: What is the definition of a logarithm: $log_b(a)$ equals what exponential statement?

Answer: $\log_b(a)=c \iff b^c=a$. A logarithm asks: what power of $b$ gives $a$?

Flashcard 24: What are the domain conditions for $\log_b(x)$ (base and argument restrictions)?

Answer: $x>0,\ b>0,\ b\ne 1$. Argument must be positive; base must be positive and not 1.

Flashcard 25: What is the inverse property: $b^{\log_b(x)}$ equals what (for $x>0$)?

Answer: $b^{\log_b(x)}=x$. Exponential and log functions cancel each other.

Flashcard 26: Find the exact value of $\log_4(\sqrt{2})$.

Answer: $\frac{1}{4}$. $\sqrt{2} = 2^{1/2} = 4^{1/4}$, so answer is $\frac{1}{4}$.

Flashcard 27: What is the definition of a logarithm in terms of an exponential equation?

Answer: $\log_b(a)=c \iff b^c=a$. A logarithm answers: "What power of $b$ gives $a$?"

Flashcard 28: Solve for $x$: $\log_4(x-1)=2$.

Answer: $x=17$. Convert to exponential: $x-1=4^2=16$, so $x=17$.

Flashcard 29: What is the change-of-base formula to rewrite $\log_b(x)$ using base $a$?

Answer: $\log_b(x)=\frac{\log_a(x)}{\log_a(b)}$. Convert any log base to another using division of logs.

Flashcard 30: What is the product property of logarithms for $\log_b(MN)$?

Answer: $\log_b(MN)=\log_b(M)+\log_b(N)$. The log of a product equals the sum of the logs.

Flashcard 31: What is the inverse relationship between $b^x$ and $\log_b(x)$?

Answer: $b^{\log_b(x)}=x$ and $\log_b(b^x)=x$. Exponential and log functions undo each other.

Flashcard 32: What is the product property of logarithms?

Answer: $\log_b(MN)=\log_b(M)+\log_b(N)$. The log of a product equals the sum of the logs.

Flashcard 33: What is the quotient property of logarithms?

Answer: $\log_b\!\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$. The log of a quotient equals the difference of the logs.

Flashcard 34: What is the value of $\log_b(b)$ for any valid base $b$?

Answer: $\log_b(b)=1$. Because $b^1=b$ for any valid base $b$.

Flashcard 35: What is the simplified value: $\log_7(7^{x-2})$ equals what?

Answer: $x-2$. Log and exponential with same base cancel.

Flashcard 36: Identify the simplified value: $\log_2(32)$ equals what?

Answer: $5$. $32 = 2^5$, so $\log_2(32) = 5$.

Flashcard 37: What is the domain of $f(x)=\log_5(2x-3)$?

Answer: $x>\frac{3}{2}$. Set $2x-3>0$ and solve for $x$.

Flashcard 38: What is the exact value of $\log_3(\sqrt{3})$?

Answer: $\frac{1}{2}$. Since $\sqrt{3}=3^{1/2}$, use the power rule.

Flashcard 39: What is the change-of-base formula for $\log_b(a)$ using base $k$?

Answer: $\log_b(a)=\frac{\log_k(a)}{\log_k(b)}$. Convert between bases by dividing logs of the same base.