Algebra Examples

2 ({log}_{3} (8) + {log}_{3} (z)) - {log}_{3} (3^{4} - 7^{2})

$2 (\log_{3} (8) + \log_{3} (z)) - \log_{3} (3^{4} - 7^{2})$

Step 1

Simplify each term.

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Step 1.1

Use the product property of logarithms,

{log}_{b} (x) + {log}_{b} (y) = {log}_{b} (x y)

$\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

2 {log}_{3} (8 z) - {log}_{3} (3^{4} - 7^{2})

$2 \log_{3} (8 z) - \log_{3} (3^{4} - 7^{2})$

Step 1.2

Simplify

2 {log}_{3} (8 z)

$2 \log_{3} (8 z)$ by moving

2

$2$ inside the logarithm.

{log}_{3} ({(8 z)}^{2}) - {log}_{3} (3^{4} - 7^{2})

$\log_{3} ({(8 z)}^{2}) - \log_{3} (3^{4} - 7^{2})$

Step 1.3

Apply the product rule to

8 z

$8 z$ .

{log}_{3} (8^{2} z^{2}) - {log}_{3} (3^{4} - 7^{2})

$\log_{3} (8^{2} z^{2}) - \log_{3} (3^{4} - 7^{2})$

Step 1.4

Raise

8

$8$ to the power of

2

$2$ .

{log}_{3} (64 z^{2}) - {log}_{3} (3^{4} - 7^{2})

$\log_{3} (64 z^{2}) - \log_{3} (3^{4} - 7^{2})$

Step 1.5

Simplify each term.

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Step 1.5.1

Raise

3

$3$ to the power of

4

$4$ .

{log}_{3} (64 z^{2}) - {log}_{3} (81 - 7^{2})

$\log_{3} (64 z^{2}) - \log_{3} (81 - 7^{2})$

Step 1.5.2

Raise

7

$7$ to the power of

2

$2$ .

{log}_{3} (64 z^{2}) - {log}_{3} (81 - 1 \cdot 49)

$\log_{3} (64 z^{2}) - \log_{3} (81 - 1 \cdot 49)$

Step 1.5.3

Multiply

- 1

$- 1$ by

49

$49$ .

{log}_{3} (64 z^{2}) - {log}_{3} (81 - 49)

$\log_{3} (64 z^{2}) - \log_{3} (81 - 49)$

{log}_{3} (64 z^{2}) - {log}_{3} (81 - 49)

$\log_{3} (64 z^{2}) - \log_{3} (81 - 49)$

Step 1.6

Subtract

49

$49$ from

81

$81$ .

{log}_{3} (64 z^{2}) - {log}_{3} (32)

$\log_{3} (64 z^{2}) - \log_{3} (32)$

{log}_{3} (64 z^{2}) - {log}_{3} (32)

$\log_{3} (64 z^{2}) - \log_{3} (32)$

Step 2

Use the quotient property of logarithms,

{log}_{b} (x) - {log}_{b} (y) = {log}_{b} (\frac{x}{y})

$\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

{log}_{3} (\frac{64 z^{2}}{32})

$\log_{3} (\frac{64 z^{2}}{32})$

Step 3

Cancel the common factor of

64

$64$ and

32

$32$ .

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Step 3.1

Factor

32

$32$ out of

64 z^{2}

$64 z^{2}$ .

{log}_{3} (\frac{32 (2 z^{2})}{32})

$\log_{3} (\frac{32 (2 z^{2})}{32})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor

32

$32$ out of

32

$32$ .

{log}_{3} (\frac{32 (2 z^{2})}{32 (1)})

$\log_{3} (\frac{32 (2 z^{2})}{32 (1)})$

Step 3.2.2

Cancel the common factor.

$\log_{3} (\frac{32 (2 z^{2})}{32 \cdot 1})$

Step 3.2.3

Rewrite the expression.

$\log_{3} (\frac{2 z^{2}}{1})$

Step 3.2.4

Divide

$2 z^{2}$ by

$1$ .

$\log_{3} (2 z^{2})$