Moderate -0.8 This is a straightforward application of logarithms to solve an exponential equation—a standard C2 technique requiring only taking logs of both sides, applying log laws, and rearranging. The large exponent (200) doesn't add conceptual difficulty, just requires careful calculator work. Below average difficulty as it's purely procedural with no problem-solving element.
Introduce logarithms throughout. Drop power on at least one side. Obtain correct linear equation (now containing no powers). Attempt solution of linear equation. Obtain \(x = 146\), or better.
OR
\((2x+1) = \log_5 5^{200}\)
\(2x + 1 = 200\log_5 5\)
Answer
Marks
M1, M1, A1, M1, A1 5
Introduce \(\log_5\) on right-hand side. Drop power of 200. Obtain correct equation. Attempt solution of linear equation. Obtain \(x = 146\), or better.
$\log_3(2^{+1}) = \log_5^{200}$
$(2x+1)\log 3 = 200\log 5$
$2x + 1 = \frac{200\log 5}{\log 3}$
$x = 146$ | M1, M1, A1, M1, A1 5 | Introduce logarithms throughout. Drop power on at least one side. Obtain correct linear equation (now containing no powers). Attempt solution of linear equation. Obtain $x = 146$, or better.
**OR**
$(2x+1) = \log_5 5^{200}$
$2x + 1 = 200\log_5 5$
| M1, M1, A1, M1, A1 5 | Introduce $\log_5$ on right-hand side. Drop power of 200. Obtain correct equation. Attempt solution of linear equation. Obtain $x = 146$, or better.
3 U se logarithms to solve the equation $3 ^ { 2 x + 1 } = 5 ^ { 200 }$, giving the value of $x$ correct to 3 significant figures.
\hfill \mbox{\textit{OCR C2 2007 Q3 [5]}}