Write as a Single Logarithm 2( log base 3 of 8+ log base 3 of z)- log base 3 of 3^4-7^2

Use the product property of logarithms, <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mi>y</mi><mo>)</mo></mrow></mstyle></math> .

Simplify <math><mstyle displaystyle="true"><mn>2</mn><msub><mi>log</mi><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><mn>8</mn><mi>z</mi><mo>)</mo></mrow></mstyle></math> by moving <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> inside the logarithm.

Apply the product rule to <math><mstyle displaystyle="true"><mn>8</mn><mi>z</mi></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mn>8</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Simplify each term.

Raise <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mn>7</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>49</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>49</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>81</mn></mstyle></math> .

Use the quotient property of logarithms, <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mi>y</mi></mrow></mfrac><mo>)</mo></mrow></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>32</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>64</mn><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Cancel the common factors.

Factor <math><mstyle displaystyle="true"><mn>32</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>32</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Divide <math><mstyle displaystyle="true"><mn>2</mn><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

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