How do you solve this logarithmic function?

Question

$(16/25)^(x+3)=(125/64)^(x-1)$

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Answer 1 · 2016-11-17T23:29:49

Please see the explanation.

Use either the natural or base 10 logarithm on both sides. (I will use the natural logarithm):

$ln((16/25)^(x+3)) = ln((125/64)^(x-1))$

Use the property of all logarithms $log_b(a^c) = (c)log_b(a)$ :

$(x+3)ln(16/25) = (x-1)ln(125/64)$

Use the distributive property on both sides:

$(x)ln(16/25) + (3)ln(16/25) = (x)ln(125/64) -ln(125/64)$

Move the x terms to the left and the constant terms to the right:

$(x)ln(16/25) - (x)ln(125/64) = - (3)ln(16/25) - ln(125/64)$

Factor out x on the left:

$(x)(ln(16/25) - ln(125/64)) = - (3)ln(16/25) - ln(125/64)$

Divide both sides by the coefficient of x:

$x = - ((3)ln(16/25) + ln(125/64))/(ln(16/25) - ln(125/64))$

Multiply the denominator by the -1 in front:

$x = ((3)ln(16/25) + ln(125/64))/(ln(125/64) - ln(16/25))$

$x = -0.6$