To solve this equation, we can start by taking the natural logarithm of both sides to eliminate the exponents:
ln(1.5^(5x-7)) = ln((2/3)^(x+1))
Using the property of logarithms that ln(a^b) = b*ln(a), we can simplify this to:
(5x-7)ln(1.5) = (x+1)ln(2/3)
Now, we can distribute the logarithms on both sides:
5xln(1.5) - 7ln(1.5) = xln(2/3) + ln(2/3)
Next, we can isolate the x terms by moving everything not containing x to the other side of the equation:
5xln(1.5) - xln(2/3) = 7ln(1.5) + ln(2/3)
Factor out x on the left side:
x(5ln(1.5) - ln(2/3)) = 7ln(1.5) + ln(2/3)
Divide both sides by (5ln(1.5) - ln(2/3)) to solve for x:
x = (7ln(1.5) + ln(2/3))/(5ln(1.5) - ln(2/3))
By simplifying the expression further, we can find the numerical value of x:
x ≈ 2.117
To solve this equation, we can start by taking the natural logarithm of both sides to eliminate the exponents:
ln(1.5^(5x-7)) = ln((2/3)^(x+1))
Using the property of logarithms that ln(a^b) = b*ln(a), we can simplify this to:
(5x-7)ln(1.5) = (x+1)ln(2/3)
Now, we can distribute the logarithms on both sides:
5xln(1.5) - 7ln(1.5) = xln(2/3) + ln(2/3)
Next, we can isolate the x terms by moving everything not containing x to the other side of the equation:
5xln(1.5) - xln(2/3) = 7ln(1.5) + ln(2/3)
Factor out x on the left side:
x(5ln(1.5) - ln(2/3)) = 7ln(1.5) + ln(2/3)
Divide both sides by (5ln(1.5) - ln(2/3)) to solve for x:
x = (7ln(1.5) + ln(2/3))/(5ln(1.5) - ln(2/3))
By simplifying the expression further, we can find the numerical value of x:
x ≈ 2.117