Standard +0.3 This is a straightforward logarithm equation requiring application of log laws (power rule) and converting to exponential form, leading to a standard quadratic. The steps are routine: simplify using 2log(x+1) = log(x+1)², rearrange to get log(2x+1) - log(x+1)² = -1, apply quotient rule, convert to exponential form, expand and solve the resulting quadratic. Slightly easier than average due to being a standard textbook exercise with well-practiced techniques.
Use law of the logarithm of a power, product or quotient
M1
Obtain a correct equation in any form, free of logs
A1
e.g. \((2x+1)/(x+1)^2 = 10^{-1}\) or \(10(2x+1)/(x+1)^2 = 10^0\) or \(x^2 + 2x + 1 = 20x + 10\)
Reduce to \(x^2 - 18x - 9 = 0\), or equivalent
A1
Solve a 3-term quadratic
M1
Obtain final answers \(x = 18.487\) and \(x = -0.487\)
A1
Must be 3 d.p. Do not allow rejection
## Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $\log_{10} 10 = 1$ | B1 | $\log_{10} 10^{-1} = -1$ |
| Use law of the logarithm of a power, product or quotient | M1 | |
| Obtain a correct equation in any form, free of logs | A1 | e.g. $(2x+1)/(x+1)^2 = 10^{-1}$ or $10(2x+1)/(x+1)^2 = 10^0$ or $x^2 + 2x + 1 = 20x + 10$ |
| Reduce to $x^2 - 18x - 9 = 0$, or equivalent | A1 | |
| Solve a 3-term quadratic | M1 | |
| Obtain final answers $x = 18.487$ and $x = -0.487$ | A1 | Must be 3 d.p. Do not allow rejection |
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