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 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 2: 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 3: What is the exact solution of $\log_3(x)=\frac{1}{2}$?

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

Flashcard 4: 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 5: 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 6: 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 7: What is the exact solution of $\ln(x)=\ln(12)$?

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

Flashcard 8: What is the exact solution of $\log_2(x-1)=3$?

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

Flashcard 9: 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 10: What is the exact solution of $4^x=2$?

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

Flashcard 11: What is $\log_7(7)$?

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

Flashcard 12: What is the exact solution of $\ln(x)=2$?

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

Flashcard 13: What is $\log_2(8)$?

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

Flashcard 14: What is $\log_3(\frac{1}{9})$?

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

Flashcard 15: What is $\log_5(1)$?

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

Flashcard 16: What is $\log_7(7)$?

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

Flashcard 17: What is $\log_4(2)$?

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

Flashcard 18: What is $\log_{\frac{1}{2}}(4)$?

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

Flashcard 19: What is $\log_{10}(0.01)$?

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

Flashcard 20: What is $\ln(1)$?

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

Flashcard 21: What is $\ln(e^3)$?

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

Flashcard 22: What is $e^{\ln(7)}$?

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

Flashcard 23: What is the exact solution of $2^x=16$?

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

Flashcard 24: What is the exact solution of $3^x=\frac{1}{27}$?

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

Flashcard 25: What is the exact solution of $10^x=0.001$?

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

Flashcard 26: What is the exact solution of $\log_5(x)=3$?

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

Flashcard 27: What is the exact solution of $\log_2(x)=-4$?

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

Flashcard 28: What is the exact solution of $\ln(x)=2$?

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

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 the exact solution of $4^x=2$?

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