SOLUTION: Find the value of a: Loga 1/25 = -2/3

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Question 889692: Find the value of a:
Loga 1/25 = -2/3

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
log%28a%2C%281%2F25%29%29%22%22=%22%22-2%2F3

Use the rule:

log%28x%2C%28y%29%29%22%22=%22%22z is equivalent to x%5Ez%22%22=%22%22y.

log%28a%2C%281%2F25%29%29%22%22=%22%22-2%2F3 is equivalent to 

matrix%282%2C1%2C%22%22%2Ca%5E%28-2%2F3%29%29%22%22=%22%22%291%2F25.

We want to make the exponent of a become 1.  To do that we raise
both sides to the -3%2F2 power, the reciprocal of the power
to which a is now raised to:

matrix%282%2C1%2C%22%22%2C%28a%5E%28-2%2F3%29%29%5E%28-3%2F2%29%29matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29%29matrix%282%2C1%2C%22%22%2C%281%2F25%29%5E%28-3%2F2%29%29

Multiply those exponents and get a to the 1 power:

matrix%282%2C1%2C%22%22%2Ca%5E1%29matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29%29matrix%282%2C1%2C%22%22%2C%281%2F25%29%5E%28-3%2F2%29%29

That's technically the answer but we need to simplify it:
We write 25 as 5%5E2.

matrix%282%2C1%2C%22%22%2Ca%29matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29%29matrix%282%2C1%2C%22%22%2C%281%2F%285%5E2%29%29%5E%28-3%2F2%29%29

We write 1%2F5%5E2 as 5%5E%28-2%29

matrix%282%2C1%2C%22%22%2Ca%29matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29%29matrix%282%2C1%2C%22%22%2C%285%5E%28-2%29%29%5E%28-3%2F2%29%29

Multiply those exponents:  %28-2%29%28-3%2F2%29%22%22=%22%223

a%22%22=%22%225%5E3

a%22%22=%22%22125

Edwin