SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

L&E, 2ADV E1 SM-Bank 3 MC

Given \(m\) and \(n\) are positive constants, which expression is equal to

\(\log _m x^5=n\)

  1. \(x=n^{\frac{m}{5}}\)
  2. \(x=m^{\frac{n}{5}}\)
  3. \(x=\dfrac{n^m}{5}\)
  4. \(x=\dfrac{m^n}{5}\)
Show Answers Only

\(\Rightarrow B\)

Show Worked Solution

\(\log _m x^5=n\)

\(\text{By definition:}\)

\(x^5\) \(=m^n\)
\(x\) \(=\left(m^n\right)^{\frac{1}{5}}\)
  \(=m^{\frac{n}{5}}\)

 
\(\Rightarrow B\)

L&E, 2ADV E1 SM-Bank 2 MC

If  `y = log_a (7x - b) + 3`, then `x` is equal to

  1. `1/7 a^(y - 3) + b`
  2. `(y - 3)/(log_a(7 - b))`
  3. `1/7 (a^(y - 3) + b)`
  4. `a^(y - 3) - b/7`
Show Answers Only

`C`

Show Worked Solution
`y – 3` `= log_a (7x – b)`
`a^(y – 3)` `= 7x – b`
`a^(y – 3) + b` `= 7x`
`:. x` `= 1/7 (a^(y – 3) + b)`

`=>   C`

L&E, 2ADV E1 2014 HSC 3 MC

What is the solution to the equation  `log_2(x-1) = 8`? 

  1. `4`
  2. `17`
  3. `65`
  4. `257`
Show Answers Only

`D`

Show Worked Solution
`log_2 (x-1)` `= 8`
`x-1` `= 2^8`
`x` `= 257`

 
`=>  D`

L&E, 2ADV E1 2008 HSC 7a

Solve  `log_e x-3/log_ex=2`   (3 marks)

Show Answers Only

`x=e^3\ \ text(or)\ \ e^-1`

Show Worked Solution
 
IMPORTANT: Students should recognise this equation as a quadratic, and the best responses substituted `log_ex` with a variable such as `X`.
`log_e x-3/(log_ex)` `=2`
`(log_ex)^2-3` `=2log_e x`
`(log_ex)^2-2log_ex-3` `=0`
   
`text(Let)\  X=log_ex`  
`:.\ X^2-2X-3` `=0`
`(X-3)(X+1)` `=0`
MARKER’S COMMENT: Many responses incorrectly stated that there is no solution to `log_ex=-1` or could not find `x` given `log_ex=3`.
`X` `=3` `\ \ \ \ \ \ \ \ \ \ ` `X` `=-1`
`log_ex` `=3` `\ \ \ \ \ \ \ \ \ \ ` `log_ex` `=-1`
`x` `=e^3` `\ \ \ \ \ \ \ \ \ \ ` `x` `=e^-1`

 

`:.x=e^3\ \ text(or)\ \ e^-1`

L&E, 2ADV E1 2009 HSC 1f

Solve the equation  `lnx=2`. Give you answer correct to four decimal places.   (2 marks) 

Show Answer Only

`7.3891`

Show Worked Solutions
MARKER’S COMMENT: Write answers to enough decimal places before rounding up.
`ln x` `=2`
`log_e x` `=2`
`x` `=e^2`
  `=7.38905…`
  `=7.3891\ \ text{(to 4 d.p.)}`