How do you solve #ln(2x^2-2)-ln9=ln80#?

If the logs are being subtracted, the numbers are being divided.

We can condense two ln terms into one.

To solve the equation ( \ln(2x^2 - 2) - \ln(9) = \ln(80) ), you can use properties of logarithms.

1. Combine the logarithms on the left side using the quotient rule, which states that ( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) ):

[ \ln\left(\frac{2x^2 - 2}{9}\right) = \ln(80) ]

2. Since the natural logarithm function is one-to-one, the arguments of the logarithms must be equal:

[ \frac{2x^2 - 2}{9} = 80 ]

3. Multiply both sides of the equation by 9 to eliminate the denominator:

[ 2x^2 - 2 = 720 ]

4. Add 2 to both sides of the equation:

[ 2x^2 = 722 ]

5. Divide both sides by 2:

[ x^2 = 361 ]

6. Take the square root of both sides:

[ x = \pm \sqrt{361} ]

[ x = \pm 19 ]

So, the solutions to the equation are ( x = 19 ) and ( x = -19 ).