Expand the Logarithmic Expression log base b of ninth root of (x^4)/(y^9x^5)

Rewrite <math><mstyle displaystyle="true"><mroot><mrow><mfrac><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow><mrow><msup><mrow><mi>y</mi></mrow><mrow><mn>9</mn></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></mfrac></mrow><mrow><mn>9</mn></mrow></mroot></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mfrac><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow><mrow><msup><mrow><mi>y</mi></mrow><mrow><mn>9</mn></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></mfrac><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>9</mn></mrow></mfrac></mrow></msup></mstyle></math> .

Expand <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mo>(</mo><mfrac><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow><mrow><msup><mrow><mi>y</mi></mrow><mrow><mn>9</mn></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></mfrac><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>9</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow></mstyle></math> by moving <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>9</mn></mrow></mfrac></mstyle></math> outside the logarithm.

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

Factor <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup></mstyle></math> out of <math><mstyle displaystyle="true"><msup><mrow><mi>y</mi></mrow><mrow><mn>9</mn></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mn>5</mn></mrow></msup></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Rewrite <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>y</mi></mrow><mrow><mn>9</mn></mrow></msup><mi>x</mi></mrow></mfrac><mo>)</mo></mrow></mstyle></math> as <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>-</mo><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>9</mn></mrow></msup><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Logarithm base <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>9</mn></mrow></msup><mi>x</mi><mo>)</mo></mrow></mstyle></math> from <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>9</mn></mrow></msup><mi>x</mi><mo>)</mo></mrow></mstyle></math> as <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>9</mn></mrow></msup><mo>)</mo></mrow><mo>+</mo><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Expand <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>9</mn></mrow></msup><mo>)</mo></mrow></mstyle></math> by moving <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> outside the logarithm.

Apply the distributive property.

Multiply <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Apply the distributive property.

Factor <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mo>-</mo><mn>9</mn><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Combine <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>9</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><msub><mi>log</mi><mrow><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

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Name | six hundred fifty-eight million four hundred sixty-three thousand eight hundred thirty-eight |
---|

- 658463838 has 16 divisors, whose sum is
**1341776016** - The reverse of 658463838 is
**838364856** - Previous prime number is
**3**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 51
- Digital Root 6

Name | one billion eight hundred seventy-eight million three hundred eighty-three thousand six hundred seventy-six |
---|

- 1878383676 has 64 divisors, whose sum is
**6216660576** - The reverse of 1878383676 is
**6763838781** - Previous prime number is
**41**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 57
- Digital Root 3

Name | two hundred sixty-one million one hundred five thousand four hundred eight |
---|

- 261105408 has 1024 divisors, whose sum is
**7435450080** - The reverse of 261105408 is
**804501162** - Previous prime number is
**9**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 27
- Digital Root 9