Question 3

Pre-U Pre-U 9794/2 2018 June — Question 3 11 marks

Exam Board	Pre-U
Module	Pre-U 9794/2 (Pre-U Mathematics Paper 2)
Year	2018
Session	June
Marks	11
Topic	Exponential Equations & Modelling
Type	Simple exponential equation solving
Difficulty	Moderate -0.8 This is a straightforward exponential equation requiring taking logarithms of both sides and rearranging algebraically. It's a standard textbook exercise with a clear method (no trial needed), making it easier than average, though not trivial since it requires careful algebraic manipulation to reach the specified form.
Spec	1.06f Laws of logarithms: addition, subtraction, power rules 1.06g Equations with exponentials: solve a^x = b

3 Solve the equation \(6 ^ { 2 x - 1 } = 3 ^ { x + 2 }\), giving your answer in the form \(x = \frac { \ln a } { \ln b }\) where \(a\) and \(b\) are integers.

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- \(\ln 6^{2x-1} = \ln 3^{x+2}\), so \((2x-1)\ln 6 = (x+2)\ln 3\) M1 Introduce ln on both sides, and drop powers. Allow any log as long as bases are consistent (and also no base specified)

- Obtain correct \((2x-1)\ln 6 = (x+2)\ln 3\) A1 Allow correct equation with any log

- \(2x\ln 6 - \ln 6 = x\ln 3 + 2\ln 3\); \(2x\ln 6 - x\ln 3 = \ln 6 + 2\ln 3\); \(x(\ln 36 - \ln 3) = \ln 6 + \ln 9\) M1 Attempt to make \(x\) the subject

- \(x\ln 12 = \ln 54\) M1 Attempt correct processes to combine logs

- \(x = \frac{\ln 54}{\ln 12}\) A1 Must now be ln not any other log. If working in log to a different base then must justify change to ln. No ISW if fraction incorrectly 'cancelled'

OR

- \(6^{2x} \times 6^{-1} = 3^x \times 9\) M1 Use index laws to split indices

- \(36^x \times 6^{-1} = 3^x \times 9\) M1 Use index law to attempt same index

- \(12^x = 54\) A1 Combine like terms to obtain \(12^x = 54\)

- \(\ln 12^x = \ln 54\); \(x\ln 12 = \ln 54\) M1 Introduce ln on both sides, and drop powers

- \(x = \frac{\ln 54}{\ln 12}\) A1 Same additional guidance as A1 in main scheme

**Question 3**

- $\ln 6^{2x-1} = \ln 3^{x+2}$, so $(2x-1)\ln 6 = (x+2)\ln 3$ **M1** Introduce ln on both sides, and drop powers. Allow any log as long as bases are consistent (and also no base specified)
- Obtain correct $(2x-1)\ln 6 = (x+2)\ln 3$ **A1** Allow correct equation with any log
- $2x\ln 6 - \ln 6 = x\ln 3 + 2\ln 3$; $2x\ln 6 - x\ln 3 = \ln 6 + 2\ln 3$; $x(\ln 36 - \ln 3) = \ln 6 + \ln 9$ **M1** Attempt to make $x$ the subject
- $x\ln 12 = \ln 54$ **M1** Attempt correct processes to combine logs
- $x = \frac{\ln 54}{\ln 12}$ **A1** Must now be ln not any other log. If working in log to a different base then must justify change to ln. No ISW if fraction incorrectly 'cancelled'

OR

- $6^{2x} \times 6^{-1} = 3^x \times 9$ **M1** Use index laws to split indices
- $36^x \times 6^{-1} = 3^x \times 9$ **M1** Use index law to attempt same index
- $12^x = 54$ **A1** Combine like terms to obtain $12^x = 54$
- $\ln 12^x = \ln 54$; $x\ln 12 = \ln 54$ **M1** Introduce ln on both sides, and drop powers
- $x = \frac{\ln 54}{\ln 12}$ **A1** Same additional guidance as A1 in main scheme

Show LaTeX source

3 Solve the equation $6 ^ { 2 x - 1 } = 3 ^ { x + 2 }$, giving your answer in the form $x = \frac { \ln a } { \ln b }$ where $a$ and $b$ are integers.

\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2018 Q3 [11]}}

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