Given that $ ext{log}_3 x = a$, find in terms of $a$,
(a) $ ext{log}_3 (9x)$
(b) $ ext{log}_3 igg( rac{x^5}{81} igg)$,
giving each answer in its simplest form - Edexcel - A-Level Maths Pure - Question 8 - 2013 - Paper 5
Question 8
Given that $ ext{log}_3 x = a$, find in terms of $a$,
(a) $ ext{log}_3 (9x)$
(b) $ ext{log}_3 igg( rac{x^5}{81} igg)$,
giving each answer in its simplest f... show full transcript
Worked Solution & Example Answer:Given that $ ext{log}_3 x = a$, find in terms of $a$,
(a) $ ext{log}_3 (9x)$
(b) $ ext{log}_3 igg( rac{x^5}{81} igg)$,
giving each answer in its simplest form - Edexcel - A-Level Maths Pure - Question 8 - 2013 - Paper 5
Step 1
Find $ ext{log}_3 (9x)$
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Answer
To find extlog3(9x), we use the property of logarithms: extlog3(9x)=extlog39+extlog3x.
Since 9=32, we have: extlog39=2.
Thus, extlog3(9x)=2+extlog3x=2+a.
Therefore, extlog3(9x)=2+a.
Step 2
Find $ ext{log}_3 igg( rac{x^5}{81} igg)$
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Answer
Using the properties of logarithms again, we have: ext{log}_3 igg( rac{x^5}{81} igg) = ext{log}_3 (x^5) - ext{log}_3 (81).
Now, since 81=34, we find: extlog3(81)=4.
This gives us: ext{log}_3 igg( rac{x^5}{81} igg) = 5 ext{log}_3 x - 4 = 5a - 4.
Thus, ext{log}_3 igg( rac{x^5}{81} igg) = 5a - 4.
Step 3
Solve for $x$
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Answer
We substitute the results from parts (a) and (b) into the equation: ext{log}_3 (9x) + ext{log}_3 igg( rac{x^5}{81} igg) = 3
This becomes: (2+a)+(5a−4)=3.
Simplifying, we have: 2+a+5a−4=3 6a−2=3 6a=5 a = rac{5}{6}.
To find x, we use: ext{log}_3 x = a = rac{5}{6},
which implies: x = 3^{ rac{5}{6}}.
Calculating x gives: xextapproximately2.498.
Thus, x=2.498 to 4 significant figures.
