Logarithmic Function Manipulation - AP Precalculus

1. Find $\text{log}_3(27)$.

   3. Since $3^3 = 27$.

2. Solve for $x$: $\text{log}_4(x) = 2$.

   $x = 16$. Convert to exponential: $4^2 = 16$.

3. State the base of the common logarithm.

   10. Base 10 logarithms are called common logarithms.

4. What is the Product Rule for logarithms?

   $\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)$. Addition corresponds to multiplication inside logs.

5. What is the definition of a logarithm?

   If $b^y = x$, then $\text{log}_b(x) = y$. A logarithm asks: what power of $b$ gives $x$?

6. Express $\text{log}_4(16)$ as a simple integer.

   2. Since $4^2 = 16$.

7. What is the value of $\text{log}_2(32)$?

   5. Since $2^5 = 32$.

8. Evaluate $\text{log}_2(1)$. What is the result?

   0. Any base raised to power 0 equals 1.

9. Simplify $\text{log}_5(1)$. What is the value?

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

10. What is the logarithmic form of the equation $10^3 = 1000$?

    $\text{log}_{10}(1000) = 3$. Convert by asking: what power of 10 gives 1000?

11. Simplify $\text{log}_b(1)$. What is the value?

    0. Any base raised to power 0 equals 1.

12. What is the value of $\text{log}_5(25)$?

    2. Since $5^2 = 25$.

13. Solve for $x$: $\text{log}_5(x) = 3$.

    $x = 125$. Convert to exponential: $5^3 = 125$.

14. Evaluate $\log_{10}(100)$.

    2. Since $10^2 = 100$.

15. What is the Quotient Rule for logarithms?

    $\text{log}_b\bigg(\frac{x}{y}\bigg) = \text{log}_b(x) - \text{log}_b(y)$. Subtraction corresponds to division inside logs.

16. Express $2 \times \text{log}_b(3)$ using a logarithm property.

    $\text{log}_b(3^2)$. Use the power rule in reverse.

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

    $y = b^x$. Logarithm and exponential functions are inverses.

18. What is the value of $\text{log}_2(8)$?

    3. Since $2^3 = 8$.

19. What is the base of the natural logarithm?

    $e$. Natural logarithms use Euler's number as base.

20. Convert $\text{log}_b(x) - \text{log}_b(y)$ using a property.

    $\text{log}_b\bigg(\frac{x}{y}\bigg)$. Use the quotient rule in reverse.

21. State the Power Rule for logarithms.

    $\text{log}_b(x^n) = n \times \text{log}_b(x)$. The exponent comes down as a coefficient.

22. Simplify $\text{log}_3(27)$ to an integer.

    3. Since $3^3 = 27$.

23. What is $\text{log}_b(0)$?

    Undefined. Cannot take logarithm of zero or negative numbers.

24. If $\text{log}_b(x) = y$, express $x$ in terms of $b$ and $y$.

    $x = b^y$. Definition of logarithm in exponential form.

25. If $\text{log}_b(8) = 3$, find $b$.

    $b = 2$. Since $2^3 = 8$.

26. What is the natural logarithm of $e$?

    1. By definition, $\ln(e) = 1$.

27. Evaluate $\text{log}_{10}(100)$.

    2. Since $10^2 = 100$.

28. Convert $10^x = 100$ to logarithmic form.

    $x = \text{log}_{10}(100)$. Ask: what power of 10 gives 100?

29. Convert $\text{log}_a(x) + \text{log}_a(y)$ using a logarithm property.

    $\text{log}_a(xy)$. Use the product rule in reverse.

30. What is the Change of Base Formula for logarithms?

    $\text{log}_b(a) = \frac{\text{log}_c(a)}{\text{log}_c(b)}$. Allows conversion between different logarithm bases.

31. Convert $\text{log}_2(32)$ to exponential form.

    $2^5 = 32$. Convert by writing as $b^y = x$ form.

32. Simplify $\text{log}_b(b^7)$. What is the value?

    7. The inverse property: $\text{log}_b(b^x) = x$.

33. Evaluate $\text{log}_{10}(1000)$.

    3. Since $10^3 = 1000$.

34. Convert $b^4 = 81$ to logarithmic form.

    $\text{log}_b(81) = 4$. Ask: what power of $b$ gives 81?

35. What is the value of $\text{log}_{10}(1)$?

    0. Any base raised to power 0 equals 1.

36. What is $\text{log}_b(b)$ equal to?

    1. Any base to the first power equals itself.

37. If $\text{log}_3(x) = 4$, what is $x$?

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

38. If $\text{log}_b(9) = 2$, what is $b$?

    $b = 3$. Since $3^2 = 9$.

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