Question

given that y/√5 = √20 ×10^-3×10^5x. and lgy- mx = c for x>0 and y>0, find the values of m and c​

Answer 1

m = 5 and c = -2

Explanation:

`let \: y = √(20) * {10}^( - 3) * {10}^(5x) \\ be \: equation \: 1`

Now

`y = √(5) ( √(20) * {10}^( - 3) * {10}^(5x) ) \\ y = √(100) * {10}^( - 3) * {10}^(5x) \\ y = 10 * {10}^( - 3) * {10}^(5x)`

`using \: rules \: of \: indices \: we \: get \\ y = {10}^(1 - 3 + 5x) \\ y = {10}^( - 2 + 5x) \\`

`now \: take \: log \: of \: both \: sides \: and \: use \: laws \: of \: logarithm \\ we \: get \\`

`log(y) = log( {10}^( - 2 + 5x) ) \\ log(y) = ( - 2 + 5x) log(10) \\ log(y) = - 2 + 5x \\ log(y) - 5x = - 2 \\ \\ comparing \: this \: to \: \\ log(y) - mx = c \\ \\ we \: get \: m \: = 5 \: and \: c = - 2`