Question

logarithmic functions
write each exponential equation in logarithmic form.

1. $7^{3}=343$

$\log_{\text{base}} 343 = \text{exponent}$
$\log_{7} 343 = \underline{\quad\quad\quad}$

1. $2^{6}=64$

$\log_{\text{base}} 64 = \text{exponent}$
$\log_{2} 64 = \underline{\quad\quad\quad}$

1. $15^{2}=225$

$\log_{\text{base}} 225 = \text{exponent}$
$\log_{15} 225 = \underline{\quad\quad\quad}$

1. $2^{3}=8$

$\underline{\quad\quad\quad}$

1. $17^{0}=1$

$\underline{\quad\quad\quad}$

1. $1^{12}=1$

$\underline{\quad\quad\quad}$

1. $4^{5}=1024$

$\underline{\quad\quad\quad}$

1. $3^{6}=729$

$\underline{\quad\quad\quad}$

1. $5^{4}=625$

$\underline{\quad\quad\quad}$

write each logarithmic equation in exponential form.

1. $\log_{4} 64 = 3$

$\log_{\text{base}} 64 = \text{exponent}$
$\underline{\quad\quad\quad} = 64$

1. $\log_{8} 512 = 3$

$\log_{\text{base}} 512 = \text{exponent}$
$\underline{\quad\quad\quad} = 512$

1. $\log_{6} 36 = 2$

$\log_{\text{base}} 36 = \text{exponent}$
$\underline{\quad\quad\quad} = 36$

1. $\log_{10} 100 = 2$

$\underline{\quad\quad\quad}$

1. $\log_{5} 125 = 3$

$\underline{\quad\quad\quad}$

1. $\log_{9} 1 = 0$

$\underline{\quad\quad\quad}$

1. $\log_{2} 128 = 7$

$\underline{\quad\quad\quad}$

1. $\log_{3} 243 = 5$

$\underline{\quad\quad\quad}$

1. $\log_{100} 1,000,000 = 3$

$\underline{\quad\quad\quad}$

evaluate by using mental math.

1. $\log 10,000$

$10^{4}=10,000$
$\log 10,000 = \underline{\quad\quad\quad}$

1. $\log 100,000$

$10^{5}=100,000$
$\log 100,000 = \underline{\quad\quad\quad}$

1. $\log 1$

$10^{0}=1$
$\log 1 = \underline{\quad\quad\quad}$

1. $\log_{2} 16$

$\underline{\quad\quad\quad}$

1. $\log_{4} 1$

$\underline{\quad\quad\quad}$

1. $\log_{9} 81$

$\underline{\quad\quad\quad}$

1. $\log_{100} 100,000,000$

$\underline{\quad\quad\quad}$

1. $\log 1,000,000,000$

$\underline{\quad\quad\quad}$

1. $\log_{3} 81$

$\underline{\quad\quad\quad}$

1. $\log_{4} 64$

$\underline{\quad\quad\quad}$

1. $\log_{5} 25$

$\underline{\quad\quad\quad}$

1. $\log 1000$

$\underline{\quad\quad\quad}$

Explanation:

Step1: Convert exponential to log

For $b^y=x$, $\log_b x=y$.

1. $\log_7 343 = 3$
2. $\log_2 64 = 6$
3. $\log_{15} 225 = 2$
4. $\log_2 8 = 3$
5. $\log_{17} 1 = 0$
6. $\log_{1} 1 = 2$
7. $\log_4 1024 = 5$
8. $\log_3 729 = 6$
9. $\log_5 625 = 4$

Step2: Convert log to exponential

For $\log_b x=y$, $b^y=x$.

1. $4^3 = 64$
2. $8^3 = 512$
3. $6^2 = 36$
4. $10^2 = 100$
5. $5^3 = 125$
6. $9^0 = 1$
7. $2^7 = 128$
8. $3^5 = 243$
9. $100^3 = 1,000,000$

Step3: Evaluate logs mentally

Use $b^y=x \implies \log_b x=y$.

1. $\log 10,000 = 4$
2. $\log 100,000 = 5$
3. $\log 1 = 0$
4. $\log_2 16 = 4$ (since $2^4=16$)
5. $\log_4 1 = 0$ (since $4^0=1$)
6. $\log_9 81 = 2$ (since $9^2=81$)
7. $\log_{100} 100,000,000 = 4$ (since $100^4=10^8=100,000,000$)
8. $\log 1,000,000,000 = 9$ (since $10^9=1,000,000,000$)
9. $\log_3 81 = 4$ (since $3^4=81$)
10. $\log_4 64 = 3$ (since $4^3=64$)
11. $\log_5 25 = 2$ (since $5^2=25$)
12. $\log 1000 = 3$ (since $10^3=1000$)

Answer:

1. $\boldsymbol{3}$
2. $\boldsymbol{6}$
3. $\boldsymbol{2}$
4. $\boldsymbol{\log_2 8 = 3}$
5. $\boldsymbol{\log_{17} 1 = 0}$
6. $\boldsymbol{\log_{1} 1 = 2}$
7. $\boldsymbol{\log_4 1024 = 5}$
8. $\boldsymbol{\log_3 729 = 6}$
9. $\boldsymbol{\log_5 625 = 4}$
10. $\boldsymbol{4^3}$
11. $\boldsymbol{8^3}$
12. $\boldsymbol{6^2}$
13. $\boldsymbol{10^2 = 100}$
14. $\boldsymbol{5^3 = 125}$
15. $\boldsymbol{9^0 = 1}$
16. $\boldsymbol{2^7 = 128}$
17. $\boldsymbol{3^5 = 243}$
18. $\boldsymbol{100^3 = 1,000,000}$
19. $\boldsymbol{4}$
20. $\boldsymbol{5}$
21. $\boldsymbol{0}$
22. $\boldsymbol{4}$
23. $\boldsymbol{0}$
24. $\boldsymbol{2}$
25. $\boldsymbol{4}$
26. $\boldsymbol{9}$
27. $\boldsymbol{4}$
28. $\boldsymbol{3}$
29. $\boldsymbol{2}$
30. $\boldsymbol{3}$