Standard +0.3 Part (i) requires applying log laws (coefficient rule, combining logs) and solving a resulting linear equation after removing logs - standard technique. Part (ii) involves recognizing a quadratic in 3^y and using the quadratic formula, then applying logs to find y - slightly more sophisticated but still a familiar pattern. Both parts are routine applications of log laws with no novel insight required, making this slightly easier than average.
10. (i) Use the laws of logarithms to solve the equation
$$3 \log _ { 8 } 2 + \log _ { 8 } ( 7 - x ) = 2 + \log _ { 8 } x$$
(ii) Using algebra, find, in terms of logarithms, the exact value of \(y\) for which
$$3 ^ { 2 y } + 3 ^ { y + 1 } = 10$$
Correct attempt to solve a quadratic of the form \(ax^2 + bx \pm 10 = 0\) (may be a letter other than \(x\) or may be \(3^y\))
\(x = 2\) or \(x = 2\) and \(-5\)
A1
Correct values
\(3^y = 2 \Rightarrow y = \log_3 2\) or \(\frac{\log 2}{\log 3}\)
dM1
Correct use of logs. Need to see \(3^y = k \Rightarrow y = \log_3 k\) or \(\frac{\log k}{\log 3}\), \(k > 0\). May be implied by awrt 0.63. Allow lg and ln for log
\(y = \log_3 2\) or \(y = \frac{\log 2}{\log 3}\)
A1
Cao (And no incorrect work using "\(-5\)"). Give BOD but penalise very sloppy notation e.g. \(\log3(2)\) for \(\log_3 2\) if necessary
Correct use of logs. Need \(9^{0.5y} = k \Rightarrow 0.5y = \log_9 k\) or \(\frac{\log k}{\log 9}, k > 0\)
\(y = 2\log_9 2\) or \(y = \frac{2\log 2}{\log 9}\)
A1
cao (and no incorrect work using "\(-5\)")
Total: 5 marks
# Question 10:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| e.g. $3\log_8 2 = \log_8 2^3$, $3\log_8 2 = \log_8 8$, $3\log_8 2 = 1$, $\log_8 2 = \frac{1}{3}$, $2 = \log_8 64$ | B1 | Demonstrates a law or property of logs on either of the constant terms |
| e.g. $\log_8(7-x) - \log_8 x = \log_8\frac{(7-x)}{x}$; $\log_8 64 + \log_8 x = \log_8 64x$; $\log_8 8 + \log_8(7-x) = \log_8 8(7-x)$ | B1 | Demonstrates the addition or subtraction law of logs on two terms, at least one of which is in terms of $x$ |
| $\log_8 8(7-x) = \log_8 64x$, $\log_8\frac{(7-x)}{x} = 1$, $\log_8\frac{(7-x)}{8x} = 0$, $\log_8\frac{8(7-x)}{x} = 2$ | M1 | Correct processing leading to one of these equations or equivalent. NB needs to be a correct equation |
| $8(7-x) = 64x$, $\frac{(7-x)}{x} = 8$, $\frac{7-x}{8x} = 1$, $\frac{8(7-x)}{x} = 64$ | A1 | Correct equation with logs removed |
| $x = \frac{7}{9}$ | A1 | Accept equivalents but must be exact e.g. $\frac{56}{72}$ or $0.777\ldots$ or $0.\dot{7}$ |
**(5 marks)**
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3^y \times 3^y + 3 \times 3^y = 10$ or $3^y(3^y + 3) = 10$ or $(3^y)^2 + 3 \times 3^y = 10$ or $x = 3^y \Rightarrow x^2 + 3x = 10$ | B1 | A correct quadratic in $x$ (or $3^y$) |
| $x^2 + 3x - 10 = 0 \Rightarrow x = \ldots$ | M1 | Correct attempt to solve a quadratic of the form $ax^2 + bx \pm 10 = 0$ (may be a letter other than $x$ or may be $3^y$) |
| $x = 2$ or $x = 2$ and $-5$ | A1 | Correct values |
| $3^y = 2 \Rightarrow y = \log_3 2$ or $\frac{\log 2}{\log 3}$ | dM1 | Correct use of logs. Need to see $3^y = k \Rightarrow y = \log_3 k$ or $\frac{\log k}{\log 3}$, $k > 0$. May be implied by awrt 0.63. Allow lg and ln for log |
| $y = \log_3 2$ or $y = \frac{\log 2}{\log 3}$ | A1 | Cao (And no incorrect work using "$-5$"). Give BOD but penalise very sloppy notation e.g. $\log3(2)$ for $\log_3 2$ if necessary |
**(5 marks) — Total 10**
## Question (ii) - Logarithmic Equation:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3^{2y} + 3^{y+1} = (3^2)^y + 3(9)^{0.5y} \Rightarrow 9^y + 3(9)^{0.5y} = 10$ | B1 | Correct quadratic in $9^{0.5y}$ |
| $x^2 + 3x - 10 = 0 \Rightarrow x = 2$ (or $-5$) | M1A1 | M1: Correct attempt to solve quadratic $ax^2 + bx - 10 = 0$; A1: Correct solution(s) |
| $9^{0.5y} = 2 \Rightarrow 0.5y = \log_9 2$ or $\frac{\log 2}{\log 9}$ | dM1 | Correct use of logs. Need $9^{0.5y} = k \Rightarrow 0.5y = \log_9 k$ or $\frac{\log k}{\log 9}, k > 0$ |
| $y = 2\log_9 2$ or $y = \frac{2\log 2}{\log 9}$ | A1 | cao (and no incorrect work using "$-5$") |
| **Total: 5 marks** | | |
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10. (i) Use the laws of logarithms to solve the equation
$$3 \log _ { 8 } 2 + \log _ { 8 } ( 7 - x ) = 2 + \log _ { 8 } x$$
(ii) Using algebra, find, in terms of logarithms, the exact value of $y$ for which
$$3 ^ { 2 y } + 3 ^ { y + 1 } = 10$$
\hfill \mbox{\textit{Edexcel C12 2018 Q10 [10]}}