Solved on Dec 19, 2023

Complete the table by finding the logarithmic or exponential forms of $\log_8(8)=1$ , $\log_8(64)=2$ , $8^3=512$ , and $\log_8(1/8)=-1$ .

STEP 1

Assumptions
1. Understanding the relationship between logarithmic and exponential forms: - The logarithmic form $\log_b(a) = c$ is equivalent to the exponential form $b^c = a$ .
2. The base of the logarithm and the base of the exponent must be the same.
3. The value inside the logarithm (the argument) is the result of the exponentiation in the exponential form.

STEP 2

We need to find the exponential form for the third row, where the logarithmic form is missing. To do this, we use the relationship between logarithms and exponents.
$\log_b(a) = c \Leftrightarrow b^c = a$

STEP 3

For the third row, we have the exponential form $8^3 = 512$ . We can convert this to logarithmic form by identifying $a = 512$ , $b = 8$ , and $c = 3$ .
$\log_8(512) = 3$

STEP 4

Now, we need to find the logarithmic form for the fourth row, where the exponential form is missing. We have the logarithmic form $\log_8(512) = 3$ .

STEP 5

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.
$8^3 = 512$

STEP 6

For the fifth row, we need to find the correct exponential form for the given logarithmic form $\log_8\left(\frac{1}{8}\right) = -1$ .

STEP 7

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.
$8^{-1} = \frac{1}{8}$

STEP 8

Now we need to correct the exponential form in the second row. We have the logarithmic form $\log_8(64) = 2$ .

STEP 9

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.
$8^2 = 64$

STEP 10

For the fifth row, we need to correct the exponential form for the given logarithmic form $\log_8\left(\frac{1}{64}\right) = -2$ .

STEP 11

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.
$8^{-2} = \frac{1}{64}$

STEP 12

Now we can complete the table with the correct logarithmic and exponential forms.
\begin{tabular}{|c|c|} \hline \begin{tabular}{c} Logarithmic \\ form \end{tabular} & \begin{tabular}{c} Exponential \\ form \end{tabular} \\ \hline $\log _{8}(8)=1$ & $8^{1}=8$ \\ \hline $\log _{8}(64)=2$ & $8^{2}=64$ \\ \hline $\log _{8}(512)=3$ & $8^{3}=512$ \\ \hline $\log _{8}(512)=3$ & $8^{3}=512$ \\ \hline $\log _{8}\left(\frac{1}{8}\right)=-1$ & $8^{-1}=\frac{1}{8}$ \\ \hline \end{tabular}

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Complete the table by finding the logarithmic or exponential forms of $\log_8(8)=1$ , $\log_8(64)=2$ , $8^3=512$ , and $\log_8(1/8)=-1$ .

STEP 1

STEP 2

We need to find the exponential form for the third row, where the logarithmic form is missing. To do this, we use the relationship between logarithms and exponents.
$\log_b(a) = c \Leftrightarrow b^c = a$

STEP 3

For the third row, we have the exponential form $8^3 = 512$ . We can convert this to logarithmic form by identifying $a = 512$ , $b = 8$ , and $c = 3$ .
$\log_8(512) = 3$

STEP 4

Now, we need to find the logarithmic form for the fourth row, where the exponential form is missing. We have the logarithmic form $\log_8(512) = 3$ .

STEP 5

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.
$8^3 = 512$

STEP 6

For the fifth row, we need to find the correct exponential form for the given logarithmic form $\log_8\left(\frac{1}{8}\right) = -1$ .

STEP 7

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.
$8^{-1} = \frac{1}{8}$

STEP 8

Now we need to correct the exponential form in the second row. We have the logarithmic form $\log_8(64) = 2$ .

STEP 9

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.
$8^2 = 64$

STEP 10

For the fifth row, we need to correct the exponential form for the given logarithmic form $\log_8\left(\frac{1}{64}\right) = -2$ .

STEP 11

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.
$8^{-2} = \frac{1}{64}$

STEP 12

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Complete the table by finding the logarithmic or exponential forms of log⁡8(8)=1\log_8(8)=1log8​(8)=1, log⁡8(64)=2\log_8(64)=2log8​(64)=2, 83=5128^3=51283=512, and log⁡8(1/8)=−1\log_8(1/8)=-1log8​(1/8)=−1.

STEP 1

STEP 2

We need to find the exponential form for the third row, where the logarithmic form is missing. To do this, we use the relationship between logarithms and exponents.log⁡b(a)=c⇔bc=a\log_b(a) = c \Leftrightarrow b^c = alogb​(a)=c⇔bc=a

STEP 3

For the third row, we have the exponential form 83=5128^3 = 51283=512. We can convert this to logarithmic form by identifying a=512a = 512a=512, b=8b = 8b=8, and c=3c = 3c=3.log⁡8(512)=3\log_8(512) = 3log8​(512)=3

STEP 4

Now, we need to find the logarithmic form for the fourth row, where the exponential form is missing. We have the logarithmic form log⁡8(512)=3\log_8(512) = 3log8​(512)=3.

STEP 5

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.83=5128^3 = 51283=512

STEP 6

For the fifth row, we need to find the correct exponential form for the given logarithmic form log⁡8(18)=−1\log_8\left(\frac{1}{8}\right) = -1log8​(81​)=−1.

STEP 7

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.8−1=188^{-1} = \frac{1}{8}8−1=81​

STEP 8

Now we need to correct the exponential form in the second row. We have the logarithmic form log⁡8(64)=2\log_8(64) = 2log8​(64)=2.

STEP 9

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.82=648^2 = 6482=64

STEP 10

For the fifth row, we need to correct the exponential form for the given logarithmic form log⁡8(164)=−2\log_8\left(\frac{1}{64}\right) = -2log8​(641​)=−2.

STEP 11

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.8−2=1648^{-2} = \frac{1}{64}8−2=641​

STEP 12

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Complete the table by finding the logarithmic or exponential forms of $\log_8(8)=1$ , $\log_8(64)=2$ , $8^3=512$ , and $\log_8(1/8)=-1$ .

We need to find the exponential form for the third row, where the logarithmic form is missing. To do this, we use the relationship between logarithms and exponents.
$\log_b(a) = c \Leftrightarrow b^c = a$

For the third row, we have the exponential form $8^3 = 512$ . We can convert this to logarithmic form by identifying $a = 512$ , $b = 8$ , and $c = 3$ .
$\log_8(512) = 3$

Now, we need to find the logarithmic form for the fourth row, where the exponential form is missing. We have the logarithmic form $\log_8(512) = 3$ .

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.
$8^3 = 512$

For the fifth row, we need to find the correct exponential form for the given logarithmic form $\log_8\left(\frac{1}{8}\right) = -1$ .

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.
$8^{-1} = \frac{1}{8}$

Now we need to correct the exponential form in the second row. We have the logarithmic form $\log_8(64) = 2$ .

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.
$8^2 = 64$

For the fifth row, we need to correct the exponential form for the given logarithmic form $\log_8\left(\frac{1}{64}\right) = -2$ .

Using the relationship between logarithms and exponents, we convert the logarithmic form to exponential form.
$8^{-2} = \frac{1}{64}$