Solve the logarithmic equation. Round to the nearest ten-thousandth if necessary.
2 log 4 – log 3 + 2 log x – 4 = 0
(1 point)
Responses
12.3308
12.3308
43.3013
43.3013
86.6025
86.6025
1875
To solve the logarithmic equation 2 log 4 - log 3 + 2 log x - 4 = 0, we can simplify it first.
Using the properties of logarithms:
2 log 4 - log 3 + 2 log x - 4 = log 4^2 - log 3 + log x^2 - 4
= log 16 - log 3 + log x^2 - 4
Now we can combine the logarithms:
log (16 * x^2) - log 3 - 4 = 0
Using the quotient rule of logarithms, we can simplify it further:
log (16 * x^2 / 3) - 4 = 0
Adding 4 to both sides:
log (16 * x^2 / 3) = 4
Converting it to exponential form:
10^4 = 16 * x^2 / 3
Rearranging the equation:
16000 = 16 * x^2 / 3
Multiplying both sides by 3:
48000 = 16 * x^2
Dividing both sides by 16:
3000 = x^2
Taking the square root of both sides:
√3000 = x
x ≈ 54.7723
Rounding to the nearest ten-thousandth:
x ≈ 54.7723 ≈ 54.7720
Therefore, the solution to the logarithmic equation is approximately 54.7720.