Question 10
Given that $\log_{10}6=0.778$, determine the value of $\log_{10}600$
Not quite. 77.800 is $0.778\times100$. Multiplying the number by 100 adds to its log. It doesn’t multiply the log: $\log_{10}600=\log_{10}6+\log_{10}100$. In 2024, 50.5% of Queensland students chose (A).
Not quite. 10.778 is $0.778+10$, as if $\log_{10}100$ were 10. But $10^2=100$, so $\log_{10}100=2$. In 2024, 9.0% of Queensland students chose (B).
That’s it. Step through the hints below to see why. In 2024, 35.4% of Queensland students chose (C).
Not quite. 1.556 is $2\times0.778$, which is $\log_{10}36$. The 2 from $100=10^2$ is added to $\log_{10}6$, not multiplied by it. In 2024, 4.9% of Queensland students chose (D).
Top line: the number, on a log scale. Bottom line: its logarithm, base 10.
600 is 6 multiplied by 10 twice. Each factor of 10 adds 1 to the log, so $\log_{10}600=0.778+2=2.778$. That’s (C).
This is the value you are given, $\log_{10}6=0.778$. Move the slider to multiply or divide by 10.
Each factor of 10 adds 1 to the log. The decimal part, 0.778, never changes. Only the whole number part changes, and it counts the factors of 10.
Dividing by 10 takes 1 off the log, so the log goes below 0.778. Here the log is , which is still 0.778 minus a whole number.
You only know the log of 6, so write 600 using a 6.
$$600=6\times100=6\times10^{2}$$
The log of a product is the sum of the logs.
$\displaystyle \log_{10}(mn)$$\displaystyle =\log_{10}m+\log_{10}n$
$\displaystyle \log_{10}600$$\displaystyle =\log_{10}6+\log_{10}100$
$10^2=100$, so $\log_{10}100=2$. Add it on.
$\displaystyle \log_{10}600=0.778+2$$\displaystyle =2.778$
Sense check: 600 is between 100 and 1000, so its log must be between 2 and 3. Only (C) is.
$$\begin{aligned}\log_{10}600&=\log_{10}(6\times100)\\&=\log_{10}6+\log_{10}100\\&=0.778+2\\&=2.778\end{aligned}$$
Multiplying by 100 adds 2 to the log, because $10^2=100$.
How Queensland students answered in 2024
More students chose (A) than the answer. From the QCAA Mathematical Methods subject report for 2024. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2024 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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