1) 8^2y = 52^4y+3
2) 2^3(2y) = 2^26(4y+3)
3) 6y = 104y+78
4) -98y = 78
5) y= -0.796
To find the value of y in each of these equations, we can follow the steps to isolate y.
1) 8^(2y) = 52^(4y+3)
Start by taking the logarithm of both sides to bring down the exponent:
log(8^(2y)) = log(52^(4y+3))
Using the property of logarithms that states log(a^b) = b * log(a), we can rewrite it as:
(2y) * log(8) = (4y+3) * log(52)
Simplify the equation:
2y * log(2^3) = (4y+3) * log(2^26)
Using the property of logarithms that states log(a^b) = b * log(a), we can further simplify it as:
6y * log(2) = (4y+3) * 26 * log(2)
Simplify the right side:
6y * log(2) = (4y+3) * 26 * log(2)
Divide both sides by 6 * log(2) to isolate y:
y = [(4y+3) * 26 * log(2)] / (6 * log(2))
Simplify further if possible using the given values of log(2) and evaluate the expression.
2) 2^(3(2y)) = 2^(26(4y+3))
We can easily solve this equation by equating the exponents, as the bases are the same:
3(2y) = 26(4y+3)
Simplify the equation:
6y = 104y + 78
Bring all the terms involving y to the left side and the constants to the right side:
6y - 104y = 78
Combine like terms:
-98y = 78
Divide both sides by -98 to solve for y:
y = 78 / -98
y = -0.796
3) 6y = 104y + 78
Bring all terms involving y to one side and constants to the other side:
-98y = 78
Divide both sides by -98 to solve for y:
y = 78 / -98
y = -0.796
4) -98y = 78
Divide both sides by -98 to solve for y:
y = 78 / -98
y = -0.796
5) Solving for y directly:
y = -0.796