To solve the exponential equation 4^(3+x) = 25 for x, we need to use logarithms. We can use any base of the logarithm to solve this equation. However, some bases may be more convenient or easier to use than others. The possible bases of the logarithm that we can use to solve this equation are:
Log base 4: If we take the logarithm of both sides of the equation with base 4, we get:
log₄(4^(3+x)) = log₄(25)
(3+x)log₄(4) = log₄(25)
3+x = log₄(25)/log₄(4)
3+x = 2.3219
x ≈ -0.6781
Log base 25: If we take the logarithm of both sides of the equation with base 25, we get:
log₂₅(4^(3+x)) = log₂₅(25)
(3+x)log₂₅(4) = 1
3+x = 1/log₂₅(4)
3+x = 1/1.3219
x ≈ -1.6781
Log base x: If we take the logarithm of both sides of the equation with base x, we get:
logₓ(4^(3+x)) = logₓ(25)
(3+x)logₓ(4) = logₓ(25)
3+x = logₓ(25)/logₓ(4)
3+x = log₄(25)/log₄(x)
x = log₄(25)/log₄(x) - 3
Log base 10: If we take the logarithm of both sides of the equation with base 10, we get:
log₁₀(4^(3+x)) = log₁₀(25)
(3+x)log₁₀(4) = log₁₀(25)
3+x = log₁₀(25)/log₁₀(4)
3+x = 1.3979
x ≈ -1.6021
Therefore, we can use any of the above logarithm bases to solve for x in the given equation.