Final answer:
To solve the logarithmic equation 2log₃x + log₃4 = 4, we use properties of logarithms to combine the terms and then exponentiate to remove the logarithm. This leads to a quadratic equation 4x^2 = 81, which simplifies to x = 4.5 when rounded to the nearest tenth.
Step-by-step explanation:
To solve the following logarithmic equation and round to the nearest tenth, 2log₃x + log₃4 = 4, we use logarithm properties and algebra.
Firstly, apply the power rule of logarithms alog₃x = log₃(x^a) which simplifies our equation to:
log₃(x^2) + log₃4 = 4
Next, apply the product rule log₃x + log₃y = log₃(xy) which results in:
log₃(4x^2) = 4
We then need to remove the logarithm by exponentiating both sides with base 3:
3^log₃(4x^2) = 3^4
This simplifies further to:
4x^2 = 81
Now, divide both sides by 4:
x^2 = 20.25
x = ±√20.25
Since we are generally interested in positive solutions for x:
x ≈ 4.5 when rounded to the nearest tenth.