Solving Logarithmic and Exponential Equations Worksheet Newer ... - Free Printable

Here are the solutions to the problems on the worksheet, broken down step-by-step.

Answer the following:

1) What is the difference between a natural logarithm and a common logarithm?
* Common Logarithm ($\log$): The base is 10. It tells you what power you need to raise 10 to get the number.
* Natural Logarithm ($\ln$): The base is $e$ (Euler's number, approx 2.718). It tells you what power you need to raise $e$ to get the number.

2) What is the inverse of an exponential?
* The inverse of an exponential function is a logarithmic function. They undo each other.

3) What is the inverse of a logarithm?
* The inverse of a logarithmic function is an exponential function.

4) Word Problem: Population of Hartford
* Formula: $s = 0.05 + 0.37 \ln P$
* Given: Speed ($s$) = 1.85. We need to find Population ($P$).
* Step 1: Plug in 1.85 for $s$.
$$1.85 = 0.05 + 0.37 \ln P$$
* Step 2: Subtract 0.05 from both sides.
$$1.80 = 0.37 \ln P$$
* Step 3: Divide by 0.37.
$$\frac{1.80}{0.37} = \ln P$$
$$4.8648... = \ln P$$
* Step 4: Convert to exponential form to solve for $P$ (base is $e$).
$$P = e^{4.8648...}$$
$$P \approx 129.6$$
* Answer: Since $P$ is in thousands, there are approximately 130 thousand people living there.

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Write each expression as a single logarithm:

5) $\ln 16 - \ln 8$
* Rule: Subtraction becomes division inside the log.
* $\ln(\frac{16}{8}) = \ln(2)$

6) $3 \ln 3 + \ln 9$
* Step 1: Move the coefficient 3 up as an exponent: $\ln(3^3) + \ln 9$.
* Step 2: Simplify $3^3 = 27$. So, $\ln 27 + \ln 9$.
* Step 3: Addition becomes multiplication inside the log: $\ln(27 \cdot 9)$.
* Step 4: $27 \cdot 9 = 243$.
* Answer: $\ln 243$ (Note: This can also be written as $\ln(3^5)$ or $5 \ln 3$).

7) $\ln z - 3 \ln x$
* Step 1: Move the 3 up: $\ln z - \ln(x^3)$.
* Step 2: Subtraction becomes division.
* Answer: $\ln(\frac{z}{x^3})$

8) $4 \ln x + 3 \ln y$
* Step 1: Move coefficients up: $\ln(x^4) + \ln(y^3)$.
* Step 2: Addition becomes multiplication.
* Answer: $\ln(x^4 y^3)$

9) $\frac{1}{2} \ln 9 - \ln 3 + \ln x$
* Step 1: Handle the fraction exponent: $\ln(9^{1/2})$. The square root of 9 is 3. So, $\ln 3$.
* Expression is now: $\ln 3 - \ln 3 + \ln x$.
* Step 2: $\ln 3 - \ln 3 = 0$.
* Answer: $\ln x$

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Evaluate without a calculator:

10) $\ln e$
* Question: $e$ to what power equals $e$? The answer is 1.
* Answer: 1

11) $\ln e^{10}$
* The natural log and $e$ cancel out, leaving just the exponent.
* Answer: 10

12) $\ln e^{\frac{1}{2}}$
* Same rule, just leave the exponent.
* Answer: $\frac{1}{2}$ (or 0.5)

13) $3 \cdot \ln e$
* $\ln e = 1$. So, $3 \cdot 1$.
* Answer: 3

14) $\log 10^5$
* Common log base is 10. Question: 10 to what power is $10^5$?
* Answer: 5

15) $\ln 1$
* Question: $e$ to what power equals 1? Any number to the power of 0 is 1.
* Answer: 0

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Solve the following logarithmic/exponential equations:

*(Note: Problems 16-30 are Exponential Equations. Problems 31-59 involve Logs and Exponentials mixed. I will group them by type for clarity.)*

Exponential Equations (Isolate the term with the variable, then take the log of both sides)

16) $2^x = 3$
* Take $\log$ of both sides: $x \log 2 = \log 3$.
* $x = \frac{\log 3}{\log 2} \approx 1.585$

17) $5^x = 81.2$
* $x = \frac{\ln 81.2}{\ln 5} \approx 2.75$

18) $3^x = 100$
* $x = \frac{\log 100}{\log 3} = \frac{2}{\log 3} \approx 4.19$

19) $-5 + 3^x = 40$
* Add 5: $3^x = 45$.
* $x = \frac{\ln 45}{\ln 3} \approx 3.46$

20) $9^{2x} = 66$
* Take $\ln$: $2x \ln 9 = \ln 66$.
* $x = \frac{\ln 66}{2 \ln 9} \approx 1.32$

21) $3^x + 1 = 5$
* Subtract 1: $3^x = 4$.
* $x = \frac{\ln 4}{\ln 3} \approx 1.26$

22) $3^{2x} + 1 = 5$
* Subtract 1: $3^{2x} = 4$.
* Take $\ln$: $2x \ln 3 = \ln 4$.
* $x = \frac{\ln 4}{2 \ln 3} \approx 0.63$

23) $3^{2-x} + 1 = 5$
* Subtract 1: $3^{2-x} = 4$.
* Take $\ln$: $(2-x)\ln 3 = \ln 4$.
* Divide by $\ln 3$: $2-x = \frac{\ln 4}{\ln 3} \approx 1.26$.
* $x = 2 - 1.26 = 0.74$

24) $\log_3(x-1) = 2$
* Convert to exponential: $3^2 = x - 1$.
* $9 = x - 1 \rightarrow x = 10$.

25) $-3 + 4^{2x} = 40$
* Add 3: $4^{2x} = 43$.
* Take $\ln$: $2x \ln 4 = \ln 43$.
* $x = \frac{\ln 43}{2 \ln 4} \approx 1.35$

26) $e^{x+1} = 10$
* Take $\ln$: $x + 1 = \ln 10$.
* $x = \ln 10 - 1 \approx 1.30$

27) $5 \cdot e^{2x} = 50$
* Divide by 5: $e^{2x} = 10$.
* Take $\ln$: $2x = \ln 10$.
* $x = \frac{\ln 10}{2} \approx 1.15$

28) $\log_5(x+4) = 2$
* Convert: $5^2 = x + 4$.
* $25 = x + 4 \rightarrow x = 21$.

29) $14^{-4x} = 36$
* Take $\ln$: $-4x \ln 14 = \ln 36$.
* $x = \frac{\ln 36}{-4 \ln 14} \approx -0.34$

30) $12^{x-2} = 20$
* Take $\ln$: $(x-2)\ln 12 = \ln 20$.
* $x - 2 = \frac{\ln 20}{\ln 12} \approx 1.20$.
* $x = 3.20$.

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Logarithmic & Mixed Equations

31) $3 \cdot \ln(2x) = 12$
* Divide by 3: $\ln(2x) = 4$.
* Exponential form: $2x = e^4$.
* $x = \frac{e^4}{2} \approx 27.30$.

32) $25^{2x+1} = 144$
* Take $\ln$: $(2x+1)\ln 25 = \ln 144$.
* $2x+1 = \frac{\ln 144}{\ln 25} \approx 1.547$.
* $2x = 0.547 \rightarrow x \approx 0.27$.

33) $2^{3x-4} = 5$
* Take $\ln$: $(3x-4)\ln 2 = \ln 5$.
* $3x-4 = \frac{\ln 5}{\ln 2} \approx 2.32$.
* $3x = 6.32 \rightarrow x \approx 2.11$.

34) $4 \cdot \ln(3x) = 8$
* Divide by 4: $\ln(3x) = 2$.
* $3x = e^2$.
* $x = \frac{e^2}{3} \approx 2.46$.

35) $\log_2(x+20) = 3$
* Convert: $2^3 = x + 20$.
* $8 = x + 20 \rightarrow x = -12$.

36) $\log x - \log 5 = 2$
* Combine logs: $\log(\frac{x}{5}) = 2$.
* Convert (base 10): $\frac{x}{5} = 10^2 = 100$.
* $x = 500$.

37) $\log x - 2 \cdot \log 5 = 1$
* Power rule: $\log x - \log(5^2) = 1 \rightarrow \log x - \log 25 = 1$.
* Quotient rule: $\log(\frac{x}{25}) = 1$.
* Convert: $\frac{x}{25} = 10^1 = 10$.
* $x = 250$.

38) $\log 5 + \log x = 2$
* Product rule: $\log(5x) = 2$.
* Convert: $5x = 10^2 = 100$.
* $x = 20$.

39) $\ln x - 2 \cdot \ln 5 = 4$
* Power rule: $\ln x - \ln(5^2) = 4 \rightarrow \ln x - \ln 25 = 4$.
* Quotient rule: $\ln(\frac{x}{25}) = 4$.
* Convert: $\frac{x}{25} = e^4$.
* $x = 25e^4 \approx 1364.9$.

40) $3 \cdot \log(2x+50) = 6$
* Divide by 3: $\log(2x+50) = 2$.
* Convert: $2x + 50 = 10^2 = 100$.
* $2x = 50 \rightarrow x = 25$.

41) $\log(2x+50)^3 = 6$
* Bring exponent down: $3 \log(2x+50) = 6$.
* Divide by 3: $\log(2x+50) = 2$.
* Same as #40: $2x+50=100 \rightarrow x=25$.

42) $\log(3x-2) = 4$
* Convert: $3x - 2 = 10^4 = 10,000$.
* $3x = 10,002 \rightarrow x = 3,334$.

43) $\log_2 x - \log_2 4 = 3$
* Quotient rule: $\log_2(\frac{x}{4}) = 3$.
* Convert: $\frac{x}{4} = 2^3 = 8$.
* $x = 32$.

44) $\log_2 4 + \log_2 x = 3$
* Simplify $\log_2 4 = 2$.
* $2 + \log_2 x = 3$.
* $\log_2 x = 1$.
* $x = 2^1 = 2$.

45) $\ln(3x-2) = 4$
* Convert: $3x - 2 = e^4$.
* $3x = e^4 + 2$.
* $x = \frac{e^4 + 2}{3} \approx 18.86$.

46) $3 \cdot e^{x+5} = 27$
* Divide by 3: $e^{x+5} = 9$.
* Take $\ln$: $x + 5 = \ln 9$.
* $x = \ln 9 - 5 \approx -2.80$.

47) $3 \cdot e^{4x} = 27$
* Divide by 3: $e^{4x} = 9$.
* Take $\ln$: $4x = \ln 9$.
* $x = \frac{\ln 9}{4} \approx 0.55$.

48) $5 \cdot e^{2x-3} - 4 = 16$
* Add 4: $5 \cdot e^{2x-3} = 20$.
* Divide by 5: $e^{2x-3} = 4$.
* Take $\ln$: $2x - 3 = \ln 4$.
* $2x = \ln 4 + 3$.
* $x = \frac{\ln 4 + 3}{2} \approx 2.19$.

49) $3 \cdot 2^x + 8 = 25$
* Subtract 8: $3 \cdot 2^x = 17$.
* Divide by 3: $2^x = \frac{17}{3}$.
* Take $\log$: $x = \frac{\log(17/3)}{\log 2} \approx 2.50$.

50) $6 \cdot \ln x = 2$
* Divide by 6: $\ln x = \frac{2}{6} = \frac{1}{3}$.
* $x = e^{1/3} \approx 1.395$.

51) $7 \cdot e^{5x+8} = 21$
* Divide by 7: $e^{5x+8} = 3$.
* Take $\ln$: $5x + 8 = \ln 3$.
* $5x = \ln 3 - 8$.
* $x = \frac{\ln 3 - 8}{5} \approx -1.38$.

52) $e^{x+6} - 5 = 1$
* Add 5: $e^{x+6} = 6$.
* Take $\ln$: $x + 6 = \ln 6$.
* $x = \ln 6 - 6 \approx -4.21$.

53) $7 \cdot \ln(2x+5) = 8$
* Divide by 7: $\ln(2x+5) = \frac{8}{7}$.
* Convert: $2x + 5 = e^{8/7}$.
* $2x = e^{8/7} - 5$.
* $x = \frac{e^{8/7} - 5}{2} \approx -1.56$.

54) $\ln(2x-1)^2 = 4$
* Bring exponent down: $2 \ln(2x-1) = 4$.
* Divide by 2: $\ln(2x-1) = 2$.
* Convert: $2x - 1 = e^2$.
* $2x = e^2 + 1$.
* $x = \frac{e^2 + 1}{2} \approx 4.19$.

55) $3 - 4 \cdot \ln(8x-1) = 12$
* Subtract 3: $-4 \cdot \ln(8x-1) = 9$.
* Divide by -4: $\ln(8x-1) = -2.25$.
* Convert: $8x - 1 = e^{-2.25}$.
* $8x = e^{-2.25} + 1$.
* $x = \frac{e^{-2.25} + 1}{8} \approx 0.135$.

56) $3 \log x - \log x = 2$
* Combine like terms: $2 \log x = 2$.
* Divide by 2: $\log x = 1$.
* $x = 10^1 = 10$.

57) $-2 + \log_3 x - \log_3 4 = 1$
* Add 2: $\log_3 x - \log_3 4 = 3$.
* Quotient rule: $\log_3(\frac{x}{4}) = 3$.
* Convert: $\frac{x}{4} = 3^3 = 27$.
* $x = 27 \cdot 4 = 108$.

58) $e^{\ln 3x} = 18$
* Rule: $e^{\ln A} = A$. So, $3x = 18$.
* $x = 6$.

59) $\log_3 2 + \log_3 x - \log_3 4 = 1$
* Combine logs: $\log_3(\frac{2x}{4}) = 1 \rightarrow \log_3(\frac{x}{2}) = 1$.
* Convert: $\frac{x}{2} = 3^1 = 3$.
* $x = 6$.

Final Answer:
See the step-by-step solutions above for all 59 problems. Key answers include:
4) ~130 thousand people
5) ln 2
6) ln 243
7) ln(z/x³)
8) ln(x⁴y³)
9) ln x
10) 1
11) 10
12) 0.5
13) 3
14) 5
15) 0
24) x = 10
28) x = 21
35) x = -12
44) x = 2
56) x = 10
58) x = 6
59) x = 6