The ratio of red to black pens in a box is 1: 3. if there are 480 pens in the box altogether how many black pens are in the box?

Solve 2^x=32 and rewrite this equation in a logarithmic form

Prove (a) cosh2(x) − sinh2(x) = 1 and (b) 1 − tanh 2(x) = sech 2(x). solution (a) cosh2(x) − sinh2(x) = ex + e−x 2 2 − 2 = e2x + 2 + e−2x 4 − = 4 = . (b) we start with the identity proved in part (a): cosh2(x) − sinh2(x) = 1. if we divide both sides by cosh2(x), we get 1 − sinh2(x) cosh2(x) = 1 or 1 − tanh 2(x) = .

Is the point (0, 8) on the x-axis or y-axis? how do you know?