Show that the equation

log y = 1.54 - .008x - .658 log x
is equivalent to
y = 34.7 (1.0186)^(-x) * x^(-.658)

take the antilog of each side

y= 10^(1.54-.008x-.658logx)

y= 10^1.54 * 10^-(.008*x) * x^.658
= 34.7((10^.008)^-x)*x^.658
you can take it from here.

remember log a+logb = log ab

and a^xy= (a^x)^y

To show that the equation log y = 1.54 - .008x - .658 log x is equivalent to y = 34.7 (1.0186)^(-x) * x^(-.658), we'll use the property of logarithms that allows us to convert between logarithmic and exponential forms.

Step 1: Start with the equation log y = 1.54 - .008x - .658 log x.

Step 2: Convert the equation to exponential form using the property log base b (x) = y is equivalent to b^y = x. In this case, the base is 10.

10^(log y) = 10^(1.54 - .008x - .658 log x)

Step 3: Simplify the right side of the equation using properties of exponents and logarithms.

y = 10^(1.54) * 10^(-.008x) * 10^(-.658 log x)

Step 4: Recall that 10^a * 10^b = 10^(a + b) and 10^(-a) = 1/10^a.

y = 10^(1.54 - .008x) * (10^(-.658))^log x

Step 5: Further simplify.

y = 10^(1.54 - .008x) * (1/10^(.658))^log x

Step 6: Recall that (1/10^a)^b = (10^(-a))^b = 10^(-a*b).

y = 10^(1.54 - .008x) * 10^(-.658 * log x)

Step 7: Apply the property a^b * a^c = a^(b + c).

y = 10^(1.54 - .008x - .658 log x)

Step 8: Notice that we have arrived at the original equation log y = 1.54 - .008x - .658 log x, proving that the two equations are indeed equivalent.

Therefore, the equation log y = 1.54 - .008x - .658 log x is equivalent to y = 10^(1.54 - .008x - .658 log x).

Note: In the final step, we used the base 10 for convenience, but the exponential form can also be written as y = (1.0186)^(-x) * x^(-.658) by replacing 10 with 1.0186. This form matches the expression y = 34.7 (1.0186)^(-x) * x^(-.658) after considering the constant term.