Logarithmic Expressions - AP Precalculus

1. What is the quotient property of logarithms for $\log_b\left(\frac{M}{N}\right)$?

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

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

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

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

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

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

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

5. What is the one-to-one property of logarithms?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

13. What is the power property of logarithms?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

32. What is the product property of logarithms?

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

33. What is the quotient property of logarithms?

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

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

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

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

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

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

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

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

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

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

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

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

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

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