Moderate -0.8 This is a straightforward logarithm manipulation question requiring students to write the equation of a line (log₁₀ y = 2x + 4) and then convert to exponential form using basic log laws. The steps are routine and mechanical with no problem-solving required, making it easier than average but not trivial since it does require understanding the relationship between logarithmic and exponential forms.
The graph of \(\log_{10} y\) against \(x\) is a straight line with gradient 2 and the intercept on the vertical axis at 4.
Write down an equation for this straight line and show that \(y = 10000 \times 100^x\). [4]
Attempt correct process to remove logs. Obtain \(y = 10^{2x} \times 10^4\) and hence \(y = 10000 \times 100^x\).
OR
Answer
Marks
Guidance
\(y = 10000 \times 100^x\); \(\log_{10} y = \log_{10}10000 + \log_{10}100^x\); \(\log_{10} y = 2x + 4\) Conclude convincingly.
M1, M1, A1
Take logs of both sides. Use one correct log rule. Obtain \(\log_{10} y = 2x + 4\). Relate to \(y = mx + c\).
$\log_{10} y = 2x + 4$ | M1, A1 | State equation of form $\log_{10} y = mx + c$. State $\log_{10} y = 2x + 4$. Base 10 must be seen, or implied by later work.
$y = 10^{2x+4} = 10^{2x} \times 10^4 = 10000 \times 100^x$ **AG** | M1, A1 [4] | Attempt correct process to remove logs. Obtain $y = 10^{2x} \times 10^4$ and hence $y = 10000 \times 100^x$.
**OR**
$y = 10000 \times 100^x$; $\log_{10} y = \log_{10}10000 + \log_{10}100^x$; $\log_{10} y = 2x + 4$ Conclude convincingly. | M1, M1, A1 | Take logs of both sides. Use one correct log rule. Obtain $\log_{10} y = 2x + 4$. Relate to $y = mx + c$.
The graph of $\log_{10} y$ against $x$ is a straight line with gradient 2 and the intercept on the vertical axis at 4.
Write down an equation for this straight line and show that $y = 10000 \times 100^x$. [4]
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2016 Q3 [4]}}