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Algebra, MET2 2019 VCAA 20 MC

The expression `log_x(y) + log_y(z)`, where `x, y` and `z` are all real numbers greater than 1, is equal to

  1. `-1/(log_y(x)) - 1/(log_z(y))`
  2. `1/(log_x(y)) + 1/(log_y(z))`
  3. `-1/(log_x(y)) - 1/(log_y(z))`
  4. `1/(log_y(x)) + 1/(log_z(y))`
  5. `log_y(x) + log_z(y)`
Show Answers Only

`D`

Show Worked Solution
`log_x(y) + log_y(z)` `= (log_y(y))/(log_y(x)) + (log_z(z))/(log_z(y))`
  `= 1/(log_y(x)) + 1/(log_z(y))`

 
`=>   D`

Algebra, MET2 2011 VCAA 22 MC

The expression

`log_c(a) + log_a(b) + log_b(c)`

is equal to

  1. `1/(log_c(a)) + 1/(log_a(b)) + 1/(log_b(c))`
  2. `1/(log_a(c)) + 1/(log_b(a)) + 1/(log_c(b))`
  3. `− 1/(log_a(b)) - 1/(log_b(c)) - 1/(log_c(a))`
  4. `1/(log_a(a)) + 1/(log_b(b)) + 1/(log_c(c))`
  5. `1/(log_c(ab)) + 1/(log_b(ac)) + 1/(log_a(cb))`
Show Answers Only

`=> B`

Show Worked Solution

`text(Solution 1)`

♦ Mean mark 45%.

`text(Using Change of Base:)`

`log_c(a) + log_a(b) + log_b(c)`

`=(log_a(a))/(log_a(c)) + (log_b(b))/(log_b(a)) + (log_c(c))/(log_c(b))`

`=1/(log_a(c)) + 1/(log_b(a)) + 1/(log_c(b))`

 
`=> B`

 

`text(Solution 2)`

`text(Let)\ \ x` `=log_c(a)`
`c^x` `=a`
`x log_a c` `=log_a a`
`x` `=1/log_a c`

 

`text(Apply similarly for the other terms.)`

`=> B`

 

`text(Solution 3)`

`text(Use technology to test the truth of each statement.)`