Worksheet for practicing algebra and trigonometry logarithm problems, including converting forms, using properties, evaluating, and solving equations.

Let's solve each section of this Logarithms Worksheet step by step and explain the solutions clearly.

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Section 1: Rewrite the exponential equation in logarithmic form

We use the rule:
If $ a^b = c $, then $ \log_a c = b $

1. $ 5^3 = 125 $
→ $ \log_5 125 = 3 $

2. $ 6^{-2} = \frac{1}{36} $
→ $ \log_6 \left(\frac{1}{36}\right) = -2 $

3. $ e^3 = 20.085 $
→ $ \ln(20.085) = 3 $ (since $ \ln = \log_e $)

4. $ e^x = 4 $
→ $ \ln 4 = x $

5. $ 8^2 = 64 $
→ $ \log_8 64 = 2 $

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Section 2: Rewrite the logarithmic equation in exponential form

Use: $ \log_a b = c $ → $ a^c = b $

1. $ \log_2 8 = x $
→ $ 2^x = 8 $

2. $ \log_5 625 = 4 $
→ $ 5^4 = 625 $

3. $ \log_x 13 = 5 $
→ $ x^5 = 13 $

4. $ \log_2 \frac{1}{8} = -3 $
→ $ 2^{-3} = \frac{1}{8} $

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Section 3: Use your calculator to evaluate each logarithm (round to four decimal places)

1. $ \log 68 $
→ $ \approx 1.8325 $

2. $ \log 100 $
→ $ \log_{10} 100 = 2.0000 $

3. $ \ln 9 $
→ $ \approx 2.1972 $

4. $ \log 10 $
→ $ \log_{10} 10 = 1.0000 $

5. $ \ln 216 $
→ $ \approx 5.3753 $

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Section 4: Use properties of logarithms to expand each expression

Use:
- $ \log_a (MN) = \log_a M + \log_a N $
- $ \log_a \left(\frac{M}{N}\right) = \log_a M - \log_a N $
- $ \log_a (M^n) = n \log_a M $

1. $ \log_2 5x $
→ $ \log_2 5 + \log_2 x $

2. $ \log_8 x^4 $
→ $ 4 \log_8 x $

3. $ \log_3 \frac{5}{x} $
→ $ \log_3 5 - \log_3 x $

4. $ \ln \sqrt{z} = \ln(z^{1/2}) $
→ $ \frac{1}{2} \ln z $

5. $ \ln z(z-1)^2 $
→ $ \ln z + \ln((z-1)^2) = \ln z + 2\ln(z-1) $

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Section 5: Use properties of logarithms to write as a single logarithm

Use:
- $ \log_a M + \log_a N = \log_a(MN) $
- $ \log_a M - \log_a N = \log_a\left(\frac{M}{N}\right) $
- $ n \log_a M = \log_a(M^n) $

1. $ \ln x + \ln 2 $
→ $ \ln(2x) $

2. $ \log_4 z - \log_4 y $
→ $ \log_4\left(\frac{z}{y}\right) $

3. $ 2 \log_2(x+4) $
→ $ \log_2((x+4)^2) $

4. $ \frac{1}{3} \log_3 5x $
→ $ \log_3((5x)^{1/3}) = \log_3 \sqrt[3]{5x} $

5. $ \log_3(x-2) - \log_3(x+2) $
→ $ \log_3\left(\frac{x-2}{x+2}\right) $

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Section 6: Given:

$ \log_x 2 = 0.3562 $, $ \log_x 3 = 0.5646 $, $ \log_x 5 = 0.8271 $
Use log rules to find:

1. $ \log_x 6 = \log_x (2 \cdot 3) = \log_x 2 + \log_x 3 $
→ $ 0.3562 + 0.5646 = 0.9208 $

2. $ \log_x \frac{3}{2} = \log_x 3 - \log_x 2 $
→ $ 0.5646 - 0.3562 = 0.2084 $

3. $ \log_x 25 = \log_x (5^2) = 2 \log_x 5 $
→ $ 2 \times 0.8271 = 1.6542 $

4. $ \log_x \sqrt{2} = \log_x (2^{1/2}) = \frac{1}{2} \log_x 2 $
→ $ \frac{1}{2} \times 0.3562 = 0.1781 $

5. $ \log_x 40 = \log_x (8 \cdot 5) = \log_x (2^3 \cdot 5) = 3\log_x 2 + \log_x 5 $
→ $ 3(0.3562) + 0.8271 = 1.0686 + 0.8271 = 1.8957 $

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Section 7: Use the definition of the logarithmic function to find $ x $

Recall: $ \log_a b = c $ means $ a^c = b $

1. $ \log_5 x = 4 $
→ $ x = 5^4 = 625 $

2. $ \log_4 2 = x $
→ $ 4^x = 2 $. But $ 4^x = (2^2)^x = 2^{2x} = 2^1 $ → $ 2x = 1 $ → $ x = \frac{1}{2} $

3. $ \log_x 25 = 2 $
→ $ x^2 = 25 $ → $ x = 5 $ (since base must be positive and ≠ 1)

4. $ \log_{10} 0.1 = x $
→ $ 10^x = 0.1 = 10^{-1} $ → $ x = -1 $

5. $ \log_4 x = 2 $
→ $ x = 4^2 = 16 $

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✔ Final Answers Summary:

#### Rewrite exponential as logarithmic:
1. $ \log_5 125 = 3 $
2. $ \log_6 \left(\frac{1}{36}\right) = -2 $
3. $ \ln 20.085 = 3 $
4. $ \ln 4 = x $
5. $ \log_8 64 = 2 $

#### Rewrite logarithmic as exponential:
1. $ 2^x = 8 $
2. $ 5^4 = 625 $
3. $ x^5 = 13 $
4. $ 2^{-3} = \frac{1}{8} $

#### Calculator evaluations:
1. $ \log 68 \approx 1.8325 $
2. $ \log 100 = 2.0000 $
3. $ \ln 9 \approx 2.1972 $
4. $ \log 10 = 1.0000 $
5. $ \ln 216 \approx 5.3753 $

#### Expand logs:
1. $ \log_2 5 + \log_2 x $
2. $ 4 \log_8 x $
3. $ \log_3 5 - \log_3 x $
4. $ \frac{1}{2} \ln z $
5. $ \ln z + 2\ln(z-1) $

#### Write as single logarithm:
1. $ \ln(2x) $
2. $ \log_4\left(\frac{z}{y}\right) $
3. $ \log_2((x+4)^2) $
4. $ \log_3 \sqrt[3]{5x} $
5. $ \log_3\left(\frac{x-2}{x+2}\right) $

#### Evaluate using given values:
1. $ \log_x 6 = 0.9208 $
2. $ \log_x \frac{3}{2} = 0.2084 $
3. $ \log_x 25 = 1.6542 $
4. $ \log_x \sqrt{2} = 0.1781 $
5. $ \log_x 40 = 1.8957 $

#### Find $ x $:
1. $ x = 625 $
2. $ x = \frac{1}{2} $
3. $ x = 5 $
4. $ x = -1 $
5. $ x = 16 $

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✔ All problems solved! Let me know if you'd like this formatted as a printable PDF or need explanations for any specific part.