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L&E, 2ADV E1 EQ-Bank 6

Solve the following equation for \(a\):

\(a^{\log_e 3}=9\)    (3 marks)

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\(a=e^2\)

Show Worked Solution
\(a^{\log_e 3}\) \(=9\)   
\(\log_e  a^{ \log_e 3}\) \(=\log_e  3^2\)
\(\log_e 3 \times \log_e a\) \(=2 \log_e  3\)
\(\log _e a\) \(=\dfrac{2 \log _e 3}{\log _e 3}\)
\(\log _e a\) \(=2\)
\(a\) \(=e^2\)

L&E, 2ADV E1 SM-Bank 4

Solve the following equation for \(x\):

\(\log _3(x-4)-\log _3 x=\dfrac{4}{3} \log _3 8\)   (3 marks)

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\(x=-\dfrac{4}{15}\)

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\(\log _3(x-4)-\log _3 x\) \(=\dfrac{4}{3} \log _3 8\)
\(\log _3\left(\dfrac{x-4}{x}\right)\) \(=\log _3 8^{\frac{4}{3}}\)
\(\log _3\left(\dfrac{x-4}{x}\right)\) \(=\log _3 16\)
\(\dfrac{x-4}{x}\) \(=16\)
\(x-4\) \(=16x\)
\(15 x\) \(=-4\)
\(x\) \(=-\dfrac{4}{15}\)

L&E, 2ADV E1 2023 HSC 8 MC

What is the solution of the equation `log _a x^3=b`, where a and b are positive constants?

  1. `x=b^(a/3)`
  2. `x=a^(b/3)`
  3. `x=b^a/3`
  4. `x=a^b/3`
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`B`

Show Worked Solution
`log_a x^3` `=b`  
`3log_a x` `=b`  
`log_a x` `=b/3`  
`:.x` `=a^(b/3)`  

 
`=>B`

L&E, 2ADV E1 2016 HSC 14e

Write  `log 2 + log 4 + log 8 + … + log 512`  in the form  `a log b`  where `a` and `b` are integers greater than `1.`  (2 marks)

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`45 log 2`

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`log 2 + log 4 + log 8 + … + log 512`

`= log 2^1 + log 2^2 + log2^3 + … + log 2^9`

`= log 2 + 2 log 2 + 3 log 2 + … + 9 log 2`

`= 45 log 2`

♦ Mean mark 40%.
 
TIP: Note that `log 2 = log_10 2`

L&E, 2ADV E1 2013 HSC 9 MC

What is the solution of   `5^x=4`?

  1. `x=(log_2 4)/5`
  2. `x=4/(log_2 5)`
  3. `x=(log_2 4)/(log_2 5)`
  4. `x=log_2(4/5)`

 

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 `C`

Show Worked Solutions
`5^x` `=4`
`log_2 5^x` `=log_2 4`
`x  log_2 5` `=log_2 4`
`:.x` `=(log_2 4)/(log_2 5)`

 
`=>C`