The equation of the line of best fit is lnM = -0.12t + 4.67. Given that M = ab^t, find the value of b.

I got as far as -0.12t + 4.67 = lna + tlnb but I don't know where to go from there.

of with -0.12t + 4.67 = lna + tlnb

-0.12t + 4.67 - lna = tlnb

ln b = (-0.12t + 4.67 - lna)/t

b = e^ ( (-0.12t + 4.67 - lna)/t )

ln M = - 0.12 t + 4.67

M = e ^ ( - 0.12 t + 4.67 )

M = e ^ ( - 0.12 t ) * e ^ 4.67

M = [ e ^ ( - 0.12 ) ] ^ t * e ^ 4.67

M = 0.88692 ^ t * 106.69774

M = 106.69774 * 0.88692 ^ t

So:

M = a * b ^ t

a = 106.69774 , b = 0.88692

Excellent! I like that.

To find the value of b, we need to rearrange the equation lnM = -0.12t + 4.67 in terms of M = ab^t.

Starting from your equation: -0.12t + 4.67 = ln(a) + tln(b), let's isolate ln(b) by moving the other terms to the other side of the equation:

ln(b) = -0.12t + 4.67 - ln(a).

Now, to eliminate the natural logarithm, we can exponentiate both sides of the equation using the property e^(ln(x)) = x. In this case, since we have ln(b) on the left side, we will use base e:

e^(ln(b)) = e^(-0.12t + 4.67 - ln(a)).

Simplifying further, we get:

b = e^(-0.12t + 4.67 - ln(a)).

So the value of b is given by this equation, where t and a are the given variables in the equation M = ab^t.

Note: Remember to substitute the appropriate values for t and a when finding the numerical value of b.

Thanks to you both :)