Moderate -0.3 Part (i) is a straightforward logarithmic equation requiring taking logs of both sides and rearranging—standard C2 material. Part (ii) involves applying log laws (bringing down the coefficient, combining logs) then solving for b, which is routine manipulation. Both parts are textbook exercises with no novel insight required, making this slightly easier than average.
13. (i) Find the value of \(x\) for which
$$4 ^ { 3 x + 2 } = 3 ^ { 600 }$$
giving your answer to 4 significant figures.
(ii) Given that
$$\log _ { a } ( 3 b - 2 ) - 2 \log _ { a } 5 = 4 , \quad a > 0 , a \neq 1 , b > \frac { 2 } { 3 }$$
find an expression for \(b\) in terms of \(a\).
\(\log 4^{3x+2} = (3x+2)\log 4\) (allow \(3x + 2\log 4\)) or \(\log 3^{600} = 600\log 3\) or \(\log_4 4^{3x+2} = 3x+2\) or \(\log_3 3^{600} = 600\) or \(3x+2 = \log_4 3^{600}\)
M1
Evidence of power law of logarithms or definition of a logarithm, independent of other working. Generally for e.g. \(\log_x y^k = k\log_x y\) or \(\log_x x^k = k\) or \(\log y^k = k\log y\) where \(x, y, k\) are any variables/numbers
\(x = \frac{1}{3}\left(\frac{600\log 3}{\log 4}-2\right)\) or \(x = \frac{600\log_4 3 - 2}{3}\) or \(x = \frac{\frac{600}{\log_3 4}-2}{3}\)
A1
Correct expression or correct value for \(x\). Must be evaluatable e.g. \(x = \frac{\log_4 3^{600}-2}{3}\) is A0. May be implied by awrt 158
## Question 13(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log 4^{3x+2} = (3x+2)\log 4$ (allow $3x + 2\log 4$) or $\log 3^{600} = 600\log 3$ or $\log_4 4^{3x+2} = 3x+2$ or $\log_3 3^{600} = 600$ or $3x+2 = \log_4 3^{600}$ | M1 | Evidence of power law of logarithms or definition of a logarithm, independent of other working. Generally for e.g. $\log_x y^k = k\log_x y$ or $\log_x x^k = k$ or $\log y^k = k\log y$ where $x, y, k$ are any variables/numbers |
| $x = \frac{1}{3}\left(\frac{600\log 3}{\log 4}-2\right)$ or $x = \frac{600\log_4 3 - 2}{3}$ or $x = \frac{\frac{600}{\log_3 4}-2}{3}$ | A1 | Correct expression or correct value for $x$. Must be evaluatable e.g. $x = \frac{\log_4 3^{600}-2}{3}$ is A0. May be implied by awrt 158 |
| $x = 157.8$ | A1 | cao (Must be this value **not** awrt) |
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## Question 13(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2\log_a 5 = \log_a 25$ or $\log_a 5^2$ | B1 | |
| $\log_a(3b-2) - \log_a 25 = \log_a\frac{(3b-2)}{25}$ or $\log_a 25 + \log_a a^4 = \log_a 25a^4$ | M1 | Correct use of subtraction or addition rule |
| $a^4 = \frac{3b-2}{25}$ | dM1 | Removes logs correctly. **Dependent on previous M** |
| $b = \frac{25a^4+2}{3}$ | A1 | cao e.g. $b = \frac{25a^4}{3}+\frac{2}{3}$ |
Special Case: $\log_a(3b-2)-\log_a 25 = \log_a\frac{25}{3b-2} \Rightarrow a^4 = \frac{25}{3b-2}$ scores B1M0dM1A0
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13. (i) Find the value of $x$ for which
$$4 ^ { 3 x + 2 } = 3 ^ { 600 }$$
giving your answer to 4 significant figures.\\
(ii) Given that
$$\log _ { a } ( 3 b - 2 ) - 2 \log _ { a } 5 = 4 , \quad a > 0 , a \neq 1 , b > \frac { 2 } { 3 }$$
find an expression for $b$ in terms of $a$.
\hfill \mbox{\textit{Edexcel C12 2018 Q13 [7]}}