solve the equation 15^4a = 81^a+2 for a. Express the answer in terms of ln3 and ln5

Answer 1

If 15^4a = 81^a+2 then taking ln at both sides and using the property ln(x^y) = yln(x), we get

4a ln(15) = (a+2)ln(81). Expanding the right-hand-side:

4a ln(15) = a ln(81) + 2 ln(81). Subtracting a ln(81) at boht sides:

4a ln(15) - a ln(81) = 2 ln(81). Factoring a:

a(4 ln(15) - ln(81)) = 2 ln(81). Solving for a:

a = 2 ln(81)/(4 ln(15) - ln(81)). Expressing 15 = 3*5 and 81 = 3⁴ and using ln(xy) = ln(x) + ln(y):

a = 8 ln(3)/(4 ln(3) + 4 ln(5) - 4 ln(3)). Simplifying:

a = 8 ln(3)/(4 ln(5)).