Two variables $x$ and $y$ are related by $y=ab^{x}$, where $a$ and $b$ are constants.
The graph of $\log_{10}y$ against $x$ is a straight line passing through the points $(0,\,1.2)$ and $(4,\,2.0)$, as shown.
Find the value of $a$ and the value of $b$, giving each answer to 3 significant figures. [5]
- for taking logs: $\log_{10}y=\log_{10}a+x\log_{10}b$M1
- for identifying $\log_{10}a$ with the intercept, $\log_{10}a=1.2$M1
- $a=15.8$ (awrt)A1
- for gradient $=\dfrac{2.0-1.2}{4}=0.2=\log_{10}b$M1
- $b=1.58$ (awrt)A1
- Answer: $a=15.8,\ b=1.58$
- Examiner tip: The gradient is $\log_{10}b$, not $b$ – remember to undo the log: $b=10^{0.2}$.