Algebra 2 Flashcards: Exponents Logarithms And Their Inverse Relationship

What this deck covers

This deck focuses on Exponents Logarithms And Their Inverse Relationship, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

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 rule for logarithms: $\log_b\left(\frac{M}{N}\right)$ equals what (for $M>0$, $N>0$)?

Answer: $\log_b(M)-\log_b(N)$. Logarithm of a quotient equals difference of logarithms.

Flashcard 2: What is the horizontal asymptote of $y=b^x$ for $b>0$ and $b\ne 1$?

Answer: $y=0$. Exponential function approaches 0 as $x$ approaches negative infinity.

Flashcard 3: What is the inverse statement that matches $a^b=c$ using logarithms (with $a>0$, $a\ne 1$, $c>0$)?

Answer: $\log_a(c)=b$. Logarithmic form: the exponent that gives $c$ when $a$ is raised to it.

Flashcard 4: What is the exact value of $\log_7(\frac{1}{49})$?

Answer: $-2$. Since $7^{-2} = \frac{1}{49}$.

Flashcard 5: What is the exact solution of $\log_2(x-1)=3$?

Answer: $x=9$. Since $2^3 = 8$, so $x - 1 = 8$.

Flashcard 6: What is the value of $\ln(e^x)$ for any real $x$?

Answer: $x$. Natural logarithm and exponential base $e$ are inverse functions.

Flashcard 7: What is the exact value of $\log_{16}(\frac{1}{2})$?

Answer: $-\frac{1}{4}$. Since $16^{-1/4} = (2^4)^{-1/4} = 2^{-1} = \frac{1}{2}$.

Flashcard 8: What is the exact solution of $\log_4(x)=0$?

Answer: $x=1$. Since $4^0 = 1$.

Flashcard 9: What is the exact solution of $\log_{10}(x)= -1$?

Answer: $x=0.1$. Since $10^{-1} = 0.1$.

Flashcard 10: What is the range of $f(x)=b^x$ for $b>0$ and $b\ne 1$?

Answer: $(0,\infty)$. Exponential functions have positive outputs for all real inputs.

Flashcard 11: What is the exact value of $\log_{27}(9)$?

Answer: $\frac{2}{3}$. Since $27^{2/3} = (3^3)^{2/3} = 3^2 = 9$.

Flashcard 12: What is the exact solution of $3^x=\frac{1}{27}$?

Answer: $x=-3$. Since $3^{-3} = \frac{1}{27}$.

Flashcard 13: What is the value of $\log_b(b^x)$ for any real $x$ and valid base $b$?

Answer: $x$. Logarithmic and exponential functions are inverses.

Flashcard 14: What is the value of $10^{\log(x)}$ for $x>0$?

Answer: $x$. Common logarithm and exponential base 10 are inverse functions.

Flashcard 15: What is the exact solution of $10^x=0.001$?

Answer: $x=-3$. Since $10^{-3} = 0.001$.

Flashcard 16: What is $\log_2(8)$?

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

Flashcard 17: What is the exact solution of $4^x=2$?

Answer: $x=\frac{1}{2}$. Since $4^{1/2} = 2$.

Flashcard 18: What is the exponential statement that matches $\log_a(c)=b$ (with $a>0$, $a\ne 1$, $c>0$)?

Answer: $a^b=c$. Exponential form: $a$ raised to the power $b$ equals $c$.

Flashcard 19: What is $\ln(1)$?

Answer: $0$. Any base raised to power 0 equals 1.

Flashcard 20: What is $\log_{\frac{1}{2}}(4)$?

Answer: $-2$. Since $(\frac{1}{2})^{-2} = 2^2 = 4$.

Flashcard 21: What are the allowed base values for $\log_b(x)$ in the real number system?

Answer: $b>0$ and $b\ne 1$. Base must be positive and not equal to 1 for valid logarithms.

Flashcard 22: What is the exact value of $\log_2(\sqrt{32})$?

Answer: $\frac{5}{2}$. Using power rule: $\log_2(32^{1/2}) = \frac{1}{2} \cdot 5 = \frac{5}{2}$.

Flashcard 23: What is the domain of $\log_b(x)$ for real outputs (assume $b>0$, $b\ne 1$)?

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

Flashcard 24: What is $\log_4(2)$?

Answer: $\frac{1}{2}$. Since $4^{1/2} = \sqrt{4} = 2$.

Flashcard 25: What is the exact solution of $\ln(x)=\ln(12)$?

Answer: $x=12$. Equal natural logarithms means equal arguments.

Flashcard 26: What is $\log_3(\frac{1}{9})$?

Answer: $-2$. Since $3^{-2} = \frac{1}{9}$.

Flashcard 27: What is the exact solution of $5^{2x}=125$?

Answer: $x=\frac{3}{2}$. Since $5^3 = 125$, so $2x = 3$.

Flashcard 28: What is the product rule for logarithms: $\log_b(MN)$ equals what (for $M>0$, $N>0$)?

Answer: $\log_b(M)+\log_b(N)$. Logarithm of a product equals sum of logarithms.

Flashcard 29: What is the exact solution of $e^x=\frac{1}{e^5}$?

Answer: $x=-5$. Since $e^{-5} = \frac{1}{e^5}$.

Flashcard 30: What is $\log_7(7)$?

Answer: $1$. Any base raised to power 1 equals itself.

Flashcard 31: What is the exact solution of $\log_2(x)=\log_2(7)$?

Answer: $x=7$. Equal logarithms with same base means equal arguments.

Flashcard 32: What is the change-of-base formula for $\log_b(x)$ using natural logs?

Answer: $\log_b(x)=\frac{\ln(x)}{\ln(b)}$. Converts logarithm to any base using natural logarithms.

Flashcard 33: What is the exact value of $\log_5(125)-\log_5(5)$?

Answer: $2$. Using quotient rule: $\log_5(\frac{125}{5}) = \log_5(25) = 2$.

Flashcard 34: What is the change-of-base formula for $\log_b(x)$ using common logs?

Answer: $\log_b(x)=\frac{\log(x)}{\log(b)}$. Converts logarithm to any base using common logarithms.

Flashcard 35: What is the value of $b^{\log_b(x)}$ for $x>0$ and valid base $b$?

Answer: $x$. Exponential and logarithmic functions are inverses.

Flashcard 36: What is $e^{\ln(7)}$?

Answer: $7$. Exponential and natural logarithm are inverse functions.

Flashcard 37: What is the exact value of $\log_3(\frac{81}{3})$?

Answer: $3$. Using quotient rule: $\log_3(\frac{81}{3}) = \log_3(27) = 3$.

Flashcard 38: What is the exact value of $\log_{\frac{1}{4}}(2)$?

Answer: $-\frac{1}{2}$. Since $(\frac{1}{4})^{-1/2} = 4^{1/2} = 2$.

Flashcard 39: What is the value of $e^{\ln(x)}$ for $x>0$?

Answer: $x$. Natural logarithm and exponential base $e$ are inverse functions.

Flashcard 40: What is the exact value of $\log_8(4)$?

Answer: $\frac{2}{3}$. Since $8^{2/3} = (2^3)^{2/3} = 2^2 = 4$.

Flashcard 41: What is $\ln(e^3)$?

Answer: $3$. Natural logarithm and exponential are inverse functions.

Flashcard 42: What is the key condition to take $\log_b$ of both sides of an equation in real numbers?

Answer: Both sides must be $>0$. Logarithm domain requires positive arguments for real outputs.

Flashcard 43: What is the inverse function of $f(x)=b^x$ (with $b>0$, $b\ne 1$) written as a log?

Answer: $f^{-1}(x)=\log_b(x)$. Exponential and logarithmic functions are inverses of each other.

Flashcard 44: What is $\log_5(1)$?

Answer: $0$. Any base raised to power 0 equals 1.

Flashcard 45: What is the value of $\log_b(1)$ for any valid base $b$?

Answer: $0$. Since $b^0 = 1$ for any valid base $b$.

Flashcard 46: What is the exact solution of $2^x=16$?

Answer: $x=4$. Since $2^4 = 16$.

Flashcard 47: What is the exact solution of $\log_5(x)=3$?

Answer: $x=125$. Since $5^3 = 125$.

Flashcard 48: What is the exact solution of $\log_3(x)-\log_3(9)=1$?

Answer: $x=27$. Using quotient rule: $\log_3(\frac{x}{9}) = 1$, so $\frac{x}{9} = 3$.

Flashcard 49: What is $\log_2(32)$?

Answer: $5$. Since $2^5 = 32$.

Flashcard 50: What is the power rule for logarithms: $\log_b(M^k)$ equals what (for $M>0$)?

Answer: $k\log_b(M)$. Logarithm of a power equals exponent times logarithm of base.

Flashcard 51: What is the vertical asymptote of $y=\log_b(x)$ for any valid base $b$?

Answer: $x=0$. Logarithm approaches negative infinity as $x$ approaches 0 from the right.

Flashcard 52: What is the exact solution of $\log_2(x)+\log_2(4)=5$?

Answer: $x=8$. Using product rule: $\log_2(4x) = 5$, so $4x = 32$.

Flashcard 53: What is the value of $\log_b(b)$ for any valid base $b$?

Answer: $1$. Since $b^1 = b$ for any valid base $b$.

Flashcard 54: What is the exact solution of $2^{x+1}=32$?

Answer: $x=4$. Since $2^5 = 32$, so $x + 1 = 5$.

Flashcard 55: What is $\log_{10}(0.01)$?

Answer: $-2$. Since $10^{-2} = 0.01$.

Flashcard 56: What is the range of $g(x)=\log_b(x)$ for a valid base $b$?

Answer: $(-\infty,\infty)$. Logarithmic functions output all real numbers for positive inputs.

Flashcard 57: What is the exact value of $\log_9(3)$?

Answer: $\frac{1}{2}$. Since $9^{1/2} = 3$, so $\log_9(3) = \frac{1}{2}$.

Flashcard 58: What is the exact solution of $\ln(x)=2$?

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

Flashcard 59: What is the exact solution of $\log_3(x^2)=2$ with $x>0$?

Answer: $x=3$. Using power rule: $2\log_3(x) = 2$, so $\log_3(x) = 1$.

Flashcard 60: What is the exact solution of $\log_2(x)=-4$?

Answer: $x=\frac{1}{16}$. Since $2^{-4} = \frac{1}{16}$.

Flashcard 61: What is the meaning of $\log_b(x)$ in words (assume $b>0$, $b\ne 1$, $x>0$)?

Answer: The exponent $y$ such that $b^y=x$. The logarithm asks: what power makes $b^y = x$?

Flashcard 62: What is the exact solution of $\log_3(x)=\frac{1}{2}$?

Answer: $x=\sqrt{3}$. Since $3^{1/2} = \sqrt{3}$.

Flashcard 63: What is the inverse function of $g(x)=\log_b(x)$ (with valid base $b$) written exponentially?

Answer: $g^{-1}(x)=b^x$. Logarithmic and exponential functions are inverses of each other.

Flashcard 64: What is the exact value of $\log_2(8\cdot 4)$ using log properties?

Answer: $5$. Using product rule: $\log_2(8) + \log_2(4) = 3 + 2 = 5$.

Flashcard 65: What is the value of $\log_{10}(10^x)$ for any real $x$?

Answer: $x$. Common logarithm and exponential base 10 are inverse functions.