Solving Exponential Equations with Logarithms - Algebra 2

1. Rewrite the solution of $a\cdot 10^{ct}=d$ using $\ln$ only.

   $t=\frac{\ln\left(\frac{d}{a}\right)}{c\ln(10)}$. Apply change-of-base formula: $\log_{10}(x) = \frac{\ln(x)}{\ln(10)}$.

2. State the change-of-base formula for rewriting $\log_b(x)$ using $\log$.

   $\log_b(x)=\frac{\log(x)}{\log(b)}$. Converts any logarithm base to common logarithm form.

3. Find $t$ in $10^{3t}=10^{12}$ and give the exact value.

   $t=4$. Since the bases are equal, equate exponents: $3t = 12$, so $t = 4$.

4. Find $t$ in $4\cdot 2^{t}=64$ and express the result as a logarithm.

   $t=\log_2(16)$. Divide by 4: $2^t = 16$, then take base-2 logarithm.

5. Find $t$ in $0.2\cdot 10^{4t}=2$ and express the result as a logarithm.

   $t=\frac{\log(10)}{4}$. Multiply by 5: $10^{4t} = 10$, then take log and divide by 4.

6. Find $t$ in $5\cdot 10^{2t}=500$ and give the exact value.

   $t=1$. Since $\log(100) = 2$ and we divide by 2, we get $t = 1$.

7. Identify the final step to isolate $t$ from $ct=\log_b\left(\frac{d}{a}\right)$.

   Divide by $c$: $t=\frac{\log_b\left(\frac{d}{a}\right)}{c}$. Divide both sides by coefficient $c$ to solve for $t$.

8. Find $t$ in $2\cdot e^{t}=10$ and express the result as a logarithm.

   $t=\ln(5)$. Divide by 2: $e^t = 5$, then take natural logarithm.

9. What is the solution for $t$ in $a\cdot 10^{ct}=d$ written using common logarithm?

   $t=\frac{\log\left(\frac{d}{a}\right)}{c}$. Divide by $a$ to get $10^{ct} = \frac{d}{a}$, then apply common log.

10. Find $t$ in $2\cdot 10^{3t}=50$ and express the result as a logarithm.

    $t=\frac{\log(25)}{3}$. Divide by 2: $10^{3t} = 25$, then take log and divide by 3.

11. Identify the correct log form of the solution to $a\cdot e^{ct}=d$.

    $t=\frac{\ln\left(\frac{d}{a}\right)}{c}$. Standard form using natural logarithm for base $e$ exponentials.

12. Rewrite the solution of $a\cdot 2^{ct}=d$ using $\ln$ only.

    $t=\frac{\ln\left(\frac{d}{a}\right)}{c\ln(2)}$. Apply change-of-base formula: $\log_2(x) = \frac{\ln(x)}{\ln(2)}$.

13. Find $t$ in $0.1\cdot 10^{t}=1$ and express the result as a logarithm.

    $t=\log(10)$. Multiply by 10: $10^t = 10$, so $t = \log(10)$.

14. Find $t$ in $0.5\cdot e^{t}=4$ and express the result as a logarithm.

    $t=\ln(8)$. Multiply by 2: $e^t = 8$, then take natural logarithm.

15. What property lets you move an exponent out front: $\log_b(x^k)=?$

    $\log_b(x^k)=k\log_b(x)$. Uses the power property to bring exponents in front.

16. Find $t$ in $1.5\cdot 2^{0.5t}=6$ and express the result as a logarithm.

    $t=\frac{\log_2(4)}{0.5}$. Divide by 1.5: $2^{0.5t} = 4$, then take log and divide by 0.5.

17. Find $t$ in $6\cdot 2^{t}=3$ and express the result as a logarithm.

    $t=\log_2\left(\frac{1}{2}\right)$. Divide by 6: $2^t = \frac{1}{2}$, then take base-2 logarithm.

18. Find $t$ in $3\cdot 2^{t}=24$ and give the exact value.

    $t=3$. Since $\log_2(8) = 3$, we get $t = 3$.

19. Identify the inverse statement that justifies taking logs: $b^{y}=x$ implies what?

    $y=\log_b(x)$. Logarithm is the inverse function of exponentiation.

20. What is the solution for $t$ in $a\cdot e^{ct}=d$ written using natural logarithm?

    $t=\frac{\ln\left(\frac{d}{a}\right)}{c}$. Divide by $a$ to get $e^{ct} = \frac{d}{a}$, then apply natural log.

21. Find $t$ in $0.01\cdot 10^{t}=100$ and express the result as a logarithm.

    $t=\log(10000)$. Multiply by 100: $10^t = 10000$, so $t = \log(10000)$.

22. Identify the correct log form of the solution to $a\cdot 2^{ct}=d$.

    $t=\frac{\log_2\left(\frac{d}{a}\right)}{c}$. Standard form using base-2 logarithm for base 2 exponentials.

23. Find $t$ in $5\cdot e^{2t}=40$ and express the result as a logarithm.

    $t=\frac{\ln(8)}{2}$. Divide by 5: $e^{2t} = 8$, then take ln and divide by 2.

24. Find $t$ in $3\cdot 2^{2t}=96$ and express the result as a logarithm.

    $t=\frac{\log_2(32)}{2}$. Divide by 3: $2^{2t} = 32$, then take log and divide by 2.

25. State the change-of-base formula for rewriting $\log_b(x)$ using $\ln$.

    $\log_b(x)=\frac{\ln(x)}{\ln(b)}$. Converts any logarithm base to natural logarithm form.

26. Find $t$ in $e^{2t}=e^{6}$ and give the exact value.

    $t=3$. Since the bases are equal, equate exponents: $2t = 6$, so $t = 3$.

27. Find $t$ in $7\cdot e^{3t}=1$ and express the result as a logarithm.

    $t=\frac{\ln\left(\frac{1}{7}\right)}{3}$. Divide by 7: $e^{3t} = \frac{1}{7}$, then take ln and divide by 3.

28. Find $t$ in $9\cdot 2^{3t}=72$ and express the result as a logarithm.

    $t=\frac{\log_2(8)}{3}$. Divide by 9: $2^{3t} = 8$, then take log and divide by 3.

29. What base restriction is required for $\log_b(x)$ when solving $a b^{ct}=d$?

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

30. Find $t$ in $3\cdot e^{0.1t}=9$ and express the result as a logarithm.

    $t=\frac{\ln(3)}{0.1}$. Divide by 3: $e^{0.1t} = 3$, then take ln and divide by 0.1.

31. Find $t$ in $8\cdot 10^{-t}=2$ and express the result as a logarithm.

    $t=\frac{\log\left(\frac{1}{4}\right)}{-1}$. Divide by 8: $10^{-t} = \frac{1}{4}$, then take log and divide by -1.

32. What is the solution for $t$ in $a\cdot 2^{ct}=d$ written using base-$2$ logarithm?

    $t=\frac{\log_2\left(\frac{d}{a}\right)}{c}$. Divide by $a$ to get $2^{ct} = \frac{d}{a}$, then apply base-2 log.

33. Find $t$ in $3\cdot 10^{2t}=300$ and express the result as a logarithm.

    $t=\frac{\log(100)}{2}$. Divide by 3: $10^{2t} = 100$, then take log and divide by 2.

34. Find $t$ in $7\cdot 10^{t}=1.4$ and express the result as a logarithm.

    $t=\log\left(\frac{1}{5}\right)$. Divide by 7: $10^t = \frac{1.4}{7} = 0.2 = \frac{1}{5}$, then take log.

35. Find $t$ in $7\cdot e^{t}=7$ and give the exact value.

    $t=0$. Since $\ln(1) = 0$ when $e^t = 1$, we get $t = 0$.

36. Rewrite the solution of $a\cdot e^{ct}=d$ using $\log$ only.

    $t=\frac{\log\left(\frac{d}{a}\right)}{c\log(e)}$. Apply change-of-base formula: $\ln(x) = \frac{\log(x)}{\log(e)}$.

37. Find $t$ in $4\cdot 10^{t}=40$ and give the exact value.

    $t=1$. Since $\log(10) = 1$, we get $t = 1$.

38. Find $t$ in $2\cdot 2^{t}=64$ and give the exact value.

    $t=5$. Since $\log_2(32) = 5$, we get $t = 5$.

39. Identify the correct log form of the solution to $a\cdot 10^{ct}=d$.

    $t=\frac{\log\left(\frac{d}{a}\right)}{c}$. Standard form using common logarithm for base 10 exponentials.

40. Find $t$ in $5\cdot 10^{t}=200$ and express the result as a logarithm.

    $t=\log(40)$. Divide by 5: $10^t = 40$, then take common logarithm.

41. What property lets you rewrite $\log_b\left(\frac{d}{a}\right)$ as a difference of logs?

    $\log_b\left(\frac{d}{a}\right)=\log_b(d)-\log_b(a)$. Uses the quotient property: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$.

42. What is the domain requirement for $\log_b(x)$ in solving exponential equations?

    $x>0$. Logarithm is only defined for positive arguments.

43. Find $t$ in $2^{4t}=2^{20}$ and give the exact value.

    $t=5$. Since the bases are equal, equate exponents: $4t = 20$, so $t = 5$.

44. Find $t$ in $0.25\cdot 2^{2t}=8$ and express the result as a logarithm.

    $t=\frac{\log_2(32)}{2}$. Multiply by 4: $2^{2t} = 32$, then take log and divide by 2.

45. Find $t$ in $0.5\cdot 2^{t}=1$ and express the result as a logarithm.

    $t=\log_2(2)$. Multiply by 2: $2^t = 2$, so $t = \log_2(2)$.

46. Find $t$ in $2\cdot e^{-t}=8$ and express the result as a logarithm.

    $t=\frac{\ln(4)}{-1}$. Divide by 2: $e^{-t} = 4$, then take ln and divide by -1.

47. Find $t$ in $10\cdot e^{-0.2t}=5$ and express the result as a logarithm.

    $t=\frac{\ln\left(\frac{1}{2}\right)}{-0.2}$. Divide by 10: $e^{-0.2t} = \frac{1}{2}$, then take ln and divide by -0.2.

48. What log step isolates $ct$ after dividing: $b^{ct}=\frac{d}{a}$?

    $ct=\log_b\left(\frac{d}{a}\right)$. Take $\log_b$ of both sides to eliminate the exponential.

49. Find $t$ in $9\cdot 2^{-t}=72$ and express the result as a logarithm.

    $t=\frac{\log_2(8)}{-1}$. Divide by 9: $2^{-t} = 8$, then take log and divide by -1.

50. Find $t$ in $16\cdot 2^{t}=1$ and express the result as a logarithm.

    $t=\log_2\left(\frac{1}{16}\right)$. Divide by 16: $2^t = \frac{1}{16}$, then take base-2 log.

51. Identify the first algebra step to solve $a b^{ct}=d$ for $t$.

    Divide: $b^{ct}=\frac{d}{a}$. Isolate the exponential term before taking logarithms.

52. Find $t$ in $12\cdot 10^{0.5t}=3$ and express the result as a logarithm.

    $t=\frac{\log\left(\frac{1}{4}\right)}{0.5}$. Divide by 12: $10^{0.5t} = \frac{1}{4}$, then take log and divide by 0.5.

53. State the solution for $t$ in $a b^{ct}=d$ written using a logarithm.

    $t=\frac{\log_b\left(\frac{d}{a}\right)}{c}$. Isolate $b^{ct}$ by dividing by $a$, then take $\log_b$ of both sides.

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