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 | one billion one hundred eight million six hundred seven thousand eight hundred eight |
---|

- 1108607808 has 4096 divisors, whose sum is
**24307402752** - The reverse of 1108607808 is
**8087068011** - Previous prime number is
**11**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 39
- Digital Root 3

Name | nine hundred twenty million four hundred twenty-five thousand six hundred eighty-seven |
---|

- 920425687 has 8 divisors, whose sum is
**943356960** - The reverse of 920425687 is
**786524029** - Previous prime number is
**8969**

- Is Prime? no
- Number parity odd
- Number length 9
- Sum of Digits 43
- Digital Root 7

Name | one billion three hundred fifty-five million nine hundred eight thousand six hundred thirty-seven |
---|

- 1355908637 has 8 divisors, whose sum is
**1388328480** - The reverse of 1355908637 is
**7368095531** - Previous prime number is
**1709**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 47
- Digital Root 2