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Condense each expression to a single logarithm: 3log3 5 - 18log3 8 The two 3's are at...

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rosey-girl | Student, Grade 12 | Valedictorian

Posted April 16, 2013 at 1:09 AM via web

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Condense each expression to a single logarithm:

3log3 5 - 18log3 8

The two 3's are at the bottom of the log.

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mariloucortez | High School Teacher | (Level 3) Adjunct Educator

Posted April 16, 2013 at 1:27 AM (Answer #1)

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Recall the properties of logarithm:
`nlog_b M = log_b M^n`
and
`log_b M + log_b N = log_b (MN)`

Apply the first property.

For `3log_3 5,`

n = 3, b = 3 and M = 5. So,

`3log_3 5 = log_3 5^3`

*`5^3 = 125`

`3log_3 5 = log_3 125`

For `18log_3 8`

n =18, b = 3 and M = 8. So,

`18log_3 8 = log_3 8^18`

*`8^18= 1.801439851 x 10^16`

`18log_3 8 = log_3 1.801439851x10^16`

Then, apply the second property. It is applicable since they have the same base of 3.

b = 3, M = 125, and N = `1.801439851x10^16`

`log_3 125 + log_3 1.801439851x10^16= log_3 (125*1.801439851x10^16)`

`=log_3 2.251799814x10^18`

Thus, the answer is `log_3 2.251799814x10^18`

         

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oldnick | Valedictorian

Posted April 17, 2013 at 1:40 AM (Answer #2)

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`3log_3(5)-18log_3(8)=log_3(5^3)(8^18)`

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