Moderate -0.3 Part (a) is a straightforward logarithm application requiring one step (x = ln7/ln5). Part (b) is a standard 'quadratic in disguise' requiring substitution y = 5^x, solving y² - 12y + 35 = 0, then converting back—a routine C2 technique with no conceptual challenges beyond pattern recognition.
4. (a) Find, to 3 significant figures, the value of \(x\) for which \(5 ^ { x } = 7\).
(b) Solve the equation \(5 ^ { 2 x } - 12 \left( 5 ^ { x } \right) + 35 = 0\).
\(x = \frac{\log 7}{\log 5}\) or \(x = \log_5 7\) (correct method up to \(x = \ldots\))
M1
\(1.21\) (must be this answer, 3 s.f.)
A1
1.21 with no working: M1 A1 (even if left as \(5^{1.21}\)). Other answers rounding to 1.2 with no working: M1 A0
Part (b)
Answer
Marks
Guidance
Answer/Working
Marks
Guidance
\((5^x - 7)(5^x - 5)\) or another variable e.g. \((y-7)(y-5)\), even \((x-7)(x-5)\)
M1 A1
M: using correct quadratic, attempt to factorise \((5^x \pm 7)(5^x \pm 5)\), or attempt quadratic formula
\(5^x = 7\) or \(5^x = 5\); \(x = 1.2\) (awrt) ft from answer to (a) if used
A1ft
Allow \(\log_5 7\) or \(\frac{\log 7}{\log 5}\) instead of 1.2 for A1ft. No marks for simply substituting decimal from (a) into equation
\(x = 1\) (allow 1.0 or 1.00 or 1.000)
B1
Do not award if \(x=1\) clearly follows from wrong working
## Question 4:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = \frac{\log 7}{\log 5}$ or $x = \log_5 7$ (correct method up to $x = \ldots$) | M1 | |
| $1.21$ (must be this answer, 3 s.f.) | A1 | 1.21 with no working: M1 A1 (even if left as $5^{1.21}$). Other answers rounding to 1.2 with no working: M1 A0 |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(5^x - 7)(5^x - 5)$ or another variable e.g. $(y-7)(y-5)$, even $(x-7)(x-5)$ | M1 A1 | M: using correct quadratic, attempt to factorise $(5^x \pm 7)(5^x \pm 5)$, or attempt quadratic formula |
| $5^x = 7$ or $5^x = 5$; $x = 1.2$ (awrt) ft from answer to (a) if used | A1ft | Allow $\log_5 7$ or $\frac{\log 7}{\log 5}$ instead of 1.2 for A1ft. No marks for simply substituting decimal from (a) into equation |
| $x = 1$ (allow 1.0 or 1.00 or 1.000) | B1 | Do not award if $x=1$ clearly follows from wrong working |
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4. (a) Find, to 3 significant figures, the value of $x$ for which $5 ^ { x } = 7$.\\
(b) Solve the equation $5 ^ { 2 x } - 12 \left( 5 ^ { x } \right) + 35 = 0$.\\
\hfill \mbox{\textit{Edexcel C2 2008 Q4 [6]}}