Logarithmic Functions - AP Precalculus

1. Simplify $\text{log}_b(1)$.

   0. Any base raised to the power 0 equals 1.

2. What is $\text{log}_{10}(100)$?

   2. Since $10^2 = 100$, the answer is 2.

3. Find $x$ if $\text{log}_3(x) = 4$.

   81. Convert to exponential: $3^4 = 81$.

4. Determine the range of $f(x) = \text{log}_b(x)$.

   All real numbers. Logarithms can output any real value as input approaches 0 or infinity.

5. What does $\text{log}_b(0)$ evaluate to?

   Undefined. Logarithm of zero does not exist in real numbers.

6. What is the definition of a logarithm?

   If $b^x = y$, then $\text{log}_b(y) = x$. This defines the inverse relationship between exponentials and logarithms.

7. What is $\text{ln}(1)$?

   0. Natural log of 1 equals 0 since $e^0 = 1$.

8. What is the inverse function of $y = \text{log}_b(x)$?

   $x = b^y$. Inverse functions undo each other's operations.

9. If $f(x) = \text{log}_b(x)$, find $f(b^4)$.

   4. Logarithm and exponential with same base cancel out.

10. State the logarithm power rule.

    $\text{log}_b(x^n) = n \times \text{log}_b(x)$. Exponents become multipliers when using logarithms.

11. What is the change of base formula?

    $\text{log}_b(x) = \frac{\text{log}_k(x)}{\text{log}_k(b)}$. Converts between different logarithm bases using any base $k$.

12. What is the value of $\text{log}_a(a)$?

    1. Any base raised to power 1 equals itself.

13. What is $\text{log}_3(1)$?

    0. Any base raised to the power 0 equals 1.

14. What is the simplified form of $\text{log}_b(b^x)$?

    $x$. Logarithm and exponential with same base cancel out.

15. Find the value of $\text{log}_7(49)$.

    2. Since $7^2 = 49$, the answer is 2.

16. Express $\text{log}_b(x^3)$ using the power rule.

    $3 \times \text{log}_b(x)$. Power rule moves the exponent as a coefficient.

17. What does $\text{ln}(e)$ equal?

    1. Natural log of its own base equals 1.

18. Convert $\text{log}_b(\frac{x}{y})$ using the quotient rule.

    $\text{log}_b(x) - \text{log}_b(y)$. Quotient rule expands division into subtraction of logs.

19. What is the base of the natural logarithm?

    The base of the natural logarithm is $e$. $e ≈ 2.718$ is Euler's number, the natural base.

20. Express $10^x = 1000$ in logarithmic form.

    $\text{log}_{10}(1000) = x$. Direct conversion from exponential to logarithmic form.

21. Evaluate $\text{log}_5(25)$.

    2. Since $5^2 = 25$, the answer is 2.

22. Identify the domain of $f(x) = \text{log}_b(x)$.

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

23. Find $\text{log}_b(b^x)$.

    x. Logarithm and exponential with same base cancel out.

24. Simplify $\text{log}_a(a^5)$.

    5. Logarithm and exponential with same base cancel out.

25. Express the exponential form of $\text{log}_5(125) = 3$.

    $5^3 = 125$. Direct conversion from logarithmic to exponential form.

26. Convert $\text{ln}(e^3)$ to a simpler form.

    3. Natural log and $e$ with same exponent cancel out.

27. What is the value of $\text{log}_b(b)$?

    1. Any base raised to the power 1 equals itself.

28. Convert the exponential equation $b^x = y$ to logarithmic form.

    $\text{log}_b(y) = x$. Direct conversion using the definition of logarithms.

29. Simplify $\text{log}_b(\frac{b^2}{b^3})$.

    -1. Quotient simplifies to $b^{-1}$, so $\text{log}_b(b^{-1}) = -1$.

30. Express $\text{log}_b(\frac{1}{x})$ using logarithm rules.

    $-\text{log}_b(x)$. Reciprocal creates negative exponent, so negative log.

31. Simplify $\log_a(a^5)$.

    5. Logarithm and exponential with same base cancel out.

32. Express $e^{\text{ln}(x)}$ in simpler form.

    x. Exponential and natural log are inverse functions.

33. What is the logarithm quotient rule?

    $\text{log}_b(\frac{x}{y}) = \text{log}_b(x) - \text{log}_b(y)$. Quotients inside logs become differences of logs.

34. Evaluate $\text{log}_{10}(0.1)$.

    -1. Since $10^{-1} = 0.1$, the answer is -1.

35. What is the base of the common logarithm?

    10. Common log uses base 10 by convention.

36. Express $\text{log}_b(a^n)$ using the power rule.

    $n \times \text{log}_b(a)$. Power rule moves exponent $n$ as coefficient.

37. What is $\text{log}_b(b^0)$?

    0. Since $b^0 = 1$ and $\text{log}_b(1) = 0$.

38. Solve for $x$: $\text{log}_4(64) = x$.

    3. Since $4^3 = 64$, the answer is 3.

39. What is the logarithm product rule?

    $\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)$. Products inside logs become sums of logs.

40. Evaluate $\text{log}_2(8)$.

    3. Since $2^3 = 8$, the answer is 3.

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