Solve the logarithmic equation. Round to the nearest ten-thousandth if necessary.2 log 4 – log 3 + 2 log x – 4 = 0(1 point)Responses12.330812.330843.301343.301386.602586.60251,875
To solve the logarithmic equation 2 log 4 – log 3 + 2 log x – 4 = 0, we can simplify it as follows:
Using the logarithmic property log(a) + log(b) = log(ab), we can combine the terms:
log(4^2) - log(3) + log(x^2) - 4 = 0
log(16) - log(3) + log(x^2) - 4 = 0
Using another logarithmic property log(a) - log(b) = log(a/b), we can simplify further:
log(16/3) + log(x^2) - 4 = 0
Using the property log(a) + log(b) = log(ab) again, we can combine the terms:
log(16/3 * x^2) - 4 = 0
Using the exponential property log(base)a = b is equivalent to a = base^b:
16/3 * x^2 = 10^4
Multiplying both sides by 3/16:
x^2 = (3/16) * 10^4
Simplifying:
x^2 = 1875
Taking the square root of both sides:
x = ± √1875
Rounding to the nearest ten-thousandth:
x ≈ ±43.3013
Therefore, the solutions to the logarithmic equation are approximately x = 43.3013 and x = -43.3013.