Logarithm properties doubt The problem is log &#x2061;<!-- ⁡ --> ( 5.64 )

ziphumulegn 2022-07-01 Answered
Logarithm properties doubt
The problem is log ( 5.64 ) 4 . According to the properties and laws of exponents,
log ( m r ) = r log ( m ). But since the exponent is outside of the parenthesis in this problem, does it solves by like 4 log ( 5.64 ) or ( log ( 5.64 ) ) 4 ? TYIA.
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Answers (2)

Sariah Glover
Answered 2022-07-02 Author has 16 answers
Your answer will not follow the law you have given. To see the difference, notice that log ( 5.63 ) represents the power of 10 needed to attain a result of 5.63, i.e. it is the number (call it a for short) that fits the equation
10 a = 5.63
Raising both sides to the fourth power gets
( 10 a ) 4 = 5.63 4
Applying exponential rules gets
10 4 a = 5.63 4
But remember that we let a = log ( 5.63 ). It follows from the above equation that, if raising 10 to the power of a gets 5.63, then raising 10 to the power of 4 a gets 5.63 4 . That is,
log ( 5.63 4 ) = 4 a = 4 log ( 5.63 )
Think about how this is different from your question, and you should clearly see why this rule won't apply for you. Your question is to solve ( log ( 5.63 ) ) 4 . Here, following order of operations, you must first calculate ( log ( 5.63 ) and then raise it to the fourth power, rather than first considering 5.63 to the fourth power and then calculating logarithm as we did above. That is to say, using our earlier substitution for a = log ( 5.63 ), your solution is a 4 instead of 4 a
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prirodnogbk
Answered 2022-07-03 Author has 6 answers
You're right that log ( a b ) = b log ( a ). Here, we have ( log ( a ) ) b indeed. This cannot be simplified easily. You could do the following, but I don't think that will help you:
( log ( a ) ) 2 = log ( a ) log ( a ) = log ( a log a )
To calculate the answer, just calculate log 5.64 and take the fourth power of that.
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