Which equation is equivalent to 3 ln 2 – ln 4 + 5 ln x = ln 3 + 2 ln 5?

ln 2x5 = ln 75
ln (5x + 2) = ln 75
ln 2x5 = ln 13
ln (5x + 4) = ln 96

3 ln 2 – ln 4 + 5 ln x = ln 3 + 2 ln 5

ln (2^3 / 4 * x^5) = ln(3 * 5^2)
ln 2x^5 = ln 75

To find the equation that is equivalent to 3 ln 2 – ln 4 + 5 ln x = ln 3 + 2 ln 5, we can start by using logarithmic properties to simplify the equation.

First, we can combine the logarithms using the property that states ln(a) + ln(b) = ln(ab):

3 ln 2 – ln 4 + 5 ln x = ln 3 + 2 ln 5
ln(2^3) – ln 4 + ln(x^5) = ln 3 + ln(5^2)
ln(8) – ln 4 + ln(x^5) = ln 3 + ln(25)

Next, we can use another logarithmic property that states ln(a) - ln(b) = ln(a/b):

ln(8/4) + ln(x^5) = ln(3) + ln(25)
ln(2) + ln(x^5) = ln(3) + ln(25)

Now we can combine the logarithms again:

ln(2x^5) = ln(3 * 25)
ln(2x^5) = ln(75)

To get the equivalent equation, we can remove the natural logarithm by exponentiating both sides of the equation with base e:

e^(ln(2x^5)) = e^(ln(75))
2x^5 = 75

Therefore, the equivalent equation to 3 ln 2 – ln 4 + 5 ln x = ln 3 + 2 ln 5 is 2x^5 = 75.