Free Printable Properties of Logarithms Worksheets - Free Printable

Problem: Solve the given logarithmic and exponential equations and evaluate expressions.

#### Part 1: Write each equation in exponential form.

1. $\log_2 64 = 6$
- The logarithmic form $\log_b a = c$ can be rewritten as $b^c = a$.
- Here, $b = 2$, $a = 64$, and $c = 6$.
- Exponential form: $2^6 = 64$.

2. $\log_4 \frac{1}{64} = -3$
- Using the same rule, $b = 4$, $a = \frac{1}{64}$, and $c = -3$.
- Exponential form: $4^{-3} = \frac{1}{64}$.

3. $\log_{10} (0.01) = -2$
- Here, $b = 10$, $a = 0.01$, and $c = -2$.
- Exponential form: $10^{-2} = 0.01$.

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#### Part 2: Write each equation in logarithmic form.

4. $2^5 = 32$
- The exponential form $b^c = a$ can be rewritten as $\log_b a = c$.
- Here, $b = 2$, $c = 5$, and $a = 32$.
- Logarithmic form: $\log_2 32 = 5$.

5. $5^{-1/2} = \frac{\sqrt{5}}{5}$
- Here, $b = 5$, $c = -\frac{1}{2}$, and $a = \frac{\sqrt{5}}{5}$.
- Logarithmic form: $\log_5 \left(\frac{\sqrt{5}}{5}\right) = -\frac{1}{2}$.

6. $10^{-1} = 0.1$
- Here, $b = 10$, $c = -1$, and $a = 0.1$.
- Logarithmic form: $\log_{10} (0.1) = -1$.

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#### Part 3: Evaluate the expression.

7. $\log_2 8$
- We need to find $x$ such that $2^x = 8$.
- Since $8 = 2^3$, we have $x = 3$.
- Answer: $\boxed{3}$.

8. $\log_8 64$
- We need to find $x$ such that $8^x = 64$.
- Since $8 = 2^3$ and $64 = 2^6$, we can rewrite $64$ as $(2^3)^2 = 8^2$.
- Thus, $x = 2$.
- Answer: $\boxed{2}$.

9. $\log_6 216$
- We need to find $x$ such that $6^x = 216$.
- Since $216 = 6^3$, we have $x = 3$.
- Answer: $\boxed{3}$.

10. $\log_7 7$
- We need to find $x$ such that $7^x = 7$.
- Clearly, $x = 1$.
- Answer: $\boxed{1}$.

11. $\log_6 1$
- We need to find $x$ such that $6^x = 1$.
- Any base raised to the power of $0$ is $1$, so $x = 0$.
- Answer: $\boxed{0}$.

12. $\log_8 \frac{1}{8}$
- We need to find $x$ such that $8^x = \frac{1}{8}$.
- Since $\frac{1}{8} = 8^{-1}$, we have $x = -1$.
- Answer: $\boxed{-1}$.

13. $\log_7 \frac{1}{49}$
- We need to find $x$ such that $7^x = \frac{1}{49}$.
- Since $\frac{1}{49} = 7^{-2}$, we have $x = -2$.
- Answer: $\boxed{-2}$.

14. $\log_9 \frac{1}{27}$
- We need to find $x$ such that $9^x = \frac{1}{27}$.
- Rewrite $9$ and $27$ in terms of base $3$: $9 = 3^2$ and $27 = 3^3$.
- Thus, $\frac{1}{27} = (3^3)^{-1} = 3^{-3}$, and $9^x = (3^2)^x = 3^{2x}$.
- Equating exponents: $3^{2x} = 3^{-3} \implies 2x = -3 \implies x = -\frac{3}{2}$.
- Answer: $\boxed{-\frac{3}{2}}$.

15. $\log_5 \sqrt{5}$
- We need to find $x$ such that $5^x = \sqrt{5}$.
- Since $\sqrt{5} = 5^{1/2}$, we have $x = \frac{1}{2}$.
- Answer: $\boxed{\frac{1}{2}}$.

16. $\log_9 3$
- We need to find $x$ such that $9^x = 3$.
- Rewrite $9$ in terms of base $3$: $9 = 3^2$.
- Thus, $9^x = (3^2)^x = 3^{2x}$, and we need $3^{2x} = 3^1$.
- Equating exponents: $2x = 1 \implies x = \frac{1}{2}$.
- Answer: $\boxed{\frac{1}{2}}$.

17. $\log_2 16$
- We need to find $x$ such that $2^x = 16$.
- Since $16 = 2^4$, we have $x = 4$.
- Answer: $\boxed{4}$.

18. $\log_{1/2} 16$
- We need to find $x$ such that $\left(\frac{1}{2}\right)^x = 16$.
- Rewrite $\frac{1}{2}$ as $2^{-1}$: $\left(2^{-1}\right)^x = 2^{-x}$.
- Thus, $2^{-x} = 16 = 2^4$.
- Equating exponents: $-x = 4 \implies x = -4$.
- Answer: $\boxed{-4}$.

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#### Part 4: Solve for $x$.

19. $\log_6 x = 2$
- Convert to exponential form: $6^2 = x$.
- Thus, $x = 36$.
- Answer: $\boxed{36}$.

20. $\log_5 x = 3$
- Convert to exponential form: $5^3 = x$.
- Thus, $x = 125$.
- Answer: $\boxed{125}$.

21. $\log_{16} x = -1$
- Convert to exponential form: $16^{-1} = x$.
- Since $16 = 2^4$, we have $16^{-1} = \frac{1}{16}$.
- Thus, $x = \frac{1}{16}$.
- Answer: $\boxed{\frac{1}{16}}$.

22. $\log_9 x = 2$
- Convert to exponential form: $9^2 = x$.
- Thus, $x = 81$.
- Answer: $\boxed{81}$.

23. $\log_{1/4} x = -2$
- Convert to exponential form: $\left(\frac{1}{4}\right)^{-2} = x$.
- Since $\frac{1}{4} = 4^{-1}$, we have $\left(4^{-1}\right)^{-2} = 4^2$.
- Thus, $x = 16$.
- Answer: $\boxed{16}$.

24. $\log_x 64 = 3$
- Convert to exponential form: $x^3 = 64$.
- Since $64 = 4^3$, we have $x = 4$.
- Answer: $\boxed{4}$.

25. $\log_x 8 = -1$
- Convert to exponential form: $x^{-1} = 8$.
- Thus, $x = \frac{1}{8}$.
- Answer: $\boxed{\frac{1}{8}}$.

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Final Answers:

1. $2^6 = 64$
2. $4^{-3} = \frac{1}{64}$
3. $10^{-2} = 0.01$
4. $\log_2 32 = 5$
5. $\log_5 \left(\frac{\sqrt{5}}{5}\right) = -\frac{1}{2}$
6. $\log_{10} (0.1) = -1$
7. $\boxed{3}$
8. $\boxed{2}$
9. $\boxed{3}$
10. $\boxed{1}$
11. $\boxed{0}$
12. $\boxed{-1}$
13. $\boxed{-2}$
14. $\boxed{-\frac{3}{2}}$
15. $\boxed{\frac{1}{2}}$
16. $\boxed{\frac{1}{2}}$
17. $\boxed{4}$
18. $\boxed{-4}$
19. $\boxed{36}$
20. $\boxed{125}$
21. $\boxed{\frac{1}{16}}$
22. $\boxed{81}$
23. $\boxed{16}$
24. $\boxed{4}$
25. $\boxed{\frac{1}{8}}$