Moderate -0.3 Part (i) is a straightforward exponential equation requiring taking logs of both sides—a routine C2 skill. Part (ii) involves standard logarithm laws (bringing terms together, dealing with fractional log) and solving a resulting quadratic, but follows predictable textbook patterns with no novel insight required. Slightly easier than average due to being a standard two-part question with well-rehearsed techniques.
8. (i) Solve
$$5 ^ { y } = 8$$
giving your answer to 3 significant figures.
(ii) Use algebra to find the values of \(x\) for which
$$\log _ { 2 } ( x + 15 ) - 4 = \frac { 1 } { 2 } \log _ { 2 } x$$
A valid attempt to factorise or solve their 3TQ to obtain \(x=\ldots\) Dependent on all previous method marks
\(x = 1, 225\)
A1
Both \(x=1\) and \(x=225\). If both are seen, ignore any other values of \(x\leq 0\) from an otherwise correct solution
# Question 8:
## Part (i):
$$5^y = 8$$
| Answer | Mark | Guidance |
|--------|------|----------|
| $y\log 5 = \log 8$ | M1 | $y\log 5 = \log 8$ or $y = \log_5 8$ |
| $y = \frac{\log 8}{\log 5} = 1.2920...$ | A1 | awrt 1.29 |
> Allow correct answer only.
## Part (ii):
$$\log_2(x+15) - 4 = \tfrac{1}{2}\log_2 x$$
| Answer | Mark | Guidance |
|--------|------|----------|
| $\log_2(x+15) - 4 = \log_2 x^{\frac{1}{2}}$ | M1 | Applies the power law of logarithms seen **at any point in their working** |
| $\log_2\left(\frac{x+15}{x^{\frac{1}{2}}}\right) = 4$ | M1 | Applies the subtraction or addition law of logarithms **at any point in their working** |
| $\left(\frac{x+15}{x^{\frac{1}{2}}}\right) = 2^4$ | M1 | Obtains a correct expression with logs removed and no errors |
| $x - 16x^{\frac{1}{2}} + 15 = 0$ or e.g. $x^2 + 225 = 226x$ | A1 | Correct **three** term quadratic in any form |
| $(\sqrt{x}-1)(\sqrt{x}-15) = 0 \Rightarrow \sqrt{x} = \ldots$ | dddM1 | A **valid** attempt to factorise or solve their three term quadratic to obtain $\sqrt{x}=\ldots$ or $x=\ldots$ Dependent on all previous method marks |
| $\{\sqrt{x} = 1, 15\}$ | | |
| $x = 1, 225$ | A1 | **Both** $x=1$ and $x=225$. If both are seen, ignore any other values of $x\leq 0$ from an otherwise correct solution |
### Alternative:
$$2\log_2(x+15) - 8 = \log_2 x$$
| Answer | Mark | Guidance |
|--------|------|----------|
| $\log_2(x+15)^2 - 8 = \log_2 x$ | M1 | Applies the power law of logarithms |
| $\log_2\left(\frac{(x+15)^2}{x}\right) = 8$ | M1 | Applies the subtraction law of logarithms |
| $\frac{(x+15)^2}{x} = 2^8$ | M1 | Obtains a correct expression with logs removed |
| $x^2 + 30x + 225 = 256x$, i.e. $x^2 - 226x + 225 = 0$ | A1 | Correct three term quadratic in any form |
| $(x-1)(x-225) = 0 \Rightarrow x = \ldots$ | dddM1 | A valid attempt to factorise or solve their 3TQ to obtain $x=\ldots$ Dependent on all previous method marks |
| $x = 1, 225$ | A1 | **Both** $x=1$ and $x=225$. If both are seen, ignore any other values of $x\leq 0$ from an otherwise correct solution |