Q: A law of the form y=abx relates x and y. From a set of readings, lg y is plotted against x to give a straight line with gradient and vertical intercept both 1.5 each. Deduce the value of a and b.
ANS; a = 120 ; b = 10
I've tried putting lg on both sides for y=ab$^x$ and got: lg y = lg a + x lg b. I guess y-intercept = lg a and gradient = lg b ? Also, I tried comparing:
lg y - 1.5x - 1.5 == lg y - lg a - (lg b)x
I can't get further.
Actually the solutions are wrong, some error occured. The answers for a and b are ;lg a = 1.5; & ;lg b = 1.5;
You've done the correct thing in taking logarithms. We have $$\ln y = \ln a + x\ln b.$$ (Note that any logarithm base will do, but just run with this.) We're given that the slope (gradient) is $1.5$. So $\ln b = 1.5$. The y-intercept is also given so $\ln a = 1.5$. The result follows.
I'm not sure why the answer key gives the answers as such. It is possible it could be a typographical error.