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math Find a if 729^a - 57=0.

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Tushar Chandra eNotes educator | Certified Educator

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We need to find a if 729^a - 57 = 0

729^a - 57 = 0

=> 729^a = 57

As the base cannot be equated we can use logarithms

log (729^a) = log 57

use the property log a^b = b*log a

=> a*log 729 = log 57

=> a = log 57 / log 729

The value of a = log 57/log 729

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giorgiana1976 | Student

We notice that we don't have common bases. We'll use logarithms to determine the variable a.

We'll take logarithms both sides:

log3 (9^3a) = log3 57

We'll use power property of logarithms:

3a*log3 9 = log3 57

But log3 9 = log3 3^2, where log3 3 = 1

log3 3^2 = 2log3 3

log3 3^2 = 2

3a*2 = log3 57

6a = log3 3*19

We'll use product property of logarithms:

6a = log3 3 + log3 19

6a = 1 + log3 19

log3 19 = lg 19/lg 3

log3 19 = 2.70

We'll divide by 6:

a = 1/6 + 2.70/6

a = 0.16 + 0.45

a = 0.61