# Question 5:

| Answer | Mark | Guidance |
|--------|------|----------|
| $(4x-1)\log_{10}2 = (5-2x)\log_{10}3$ | M1* | Introduce logs throughout and drop power(s). Allow no base or base other than 10 as long as consistent, including $\log_3$ on LHS or $\log_2$ on RHS. Drop single power if $\log_3$ or $\log_2$ or both powers if any other base |
| $(4x-1)\log_{10}2 = (5-2x)\log_{10}3$ | A1 | Obtain $(4x-1)\log_{10}2 = (5-2x)\log_{10}3$. Brackets must be seen or implied by later working. Allow no base or base other than 10 if consistent. Any correct linear equation i.e. $4x - 1 = (5-2x)\log_2 3$ or $(4x-1)\log_3 2 = 5-2x$ |
| $x(4\log_{10}2 + 2\log_{10}3) = \log_{10}2 + 5\log_{10}3$ | M1* | Attempt to make $x$ the subject. Expand bracket(s) and collect like terms. Expressions could include $\log_2 3$ or $\log_3 2$. Must be working exactly, so M0 if $\log(s)$ now decimal equivs |
| | A1 | Obtain a correct equation in which $x$ only appears once. LHS could be $x(4\log_{10}2 + 2\log_{10}3)$, $x\log_{10}144$ or even $\log_{10}144^x$. Expressions could include $\log_2 3$ or $\log_3 2$. RHS may be two terms or single term |
| $x\log_{10}144 = \log_{10}486$ | M1d* | Attempt correct processes to combine logs. Use $b\log a = \log a^b$, then $\log a + \log b = \log ab$ correctly on at least one side of equation (or $\log a - \log b$). Dependent on previous M1 but not the A1 so $\log_{10}486$ will get this M1 irrespective of the LHS |
| $x = \frac{\log_{10}486}{\log_{10}144}$ | A1 | Obtain correct final expression. Base 10 required in final answer — allow A1 if no base earlier, or if base 10 omitted at times, but A0 if different base seen previously (unless legitimate working to change base seen). Do not allow subsequent incorrect log work e.g. $x = \frac{\log 27}{\log 8}$ |

# Mark Scheme Extraction

## Alternative Solution (Logarithms Question):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Use index laws to split both terms: $2^{4x} \div 2 = 3^5 \div 3^{2x}$, then $2^{4x} \times 3^{2x} = 3^5 \times 2$ | M1 | Either into fractions, or into products involving negative indices ie $2^{4x} \times 2^{-1}$ |
| Obtain $2^{4x} \times 3^{2x} = 3^5 \times 2$, use $a^{bx} = (a^b)^x$ giving $16^x \times 9^x = 243 \times 2$ | A1 | Combine like terms on each side |
| $144^x = 486$, so $\log_{10} 144^x = \log_{10} 486$ | M1 | Use at least once correctly |
| $x = \frac{\log_{10} 486}{\log_{10} 144}$ | A1 | Any correct equation in which $x$ appears only once; logs may have been introduced prior to this |
| Introduce logs on both sides and drop power | M1 | Allow no base, or base other than 10 if consistent |
| Obtain correct final answer | A1 | Do not isw subsequent incorrect log work |

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