Question

How do you solve this logarithmic function?

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

Answer 1

Please see the explanation.

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`