smc-963-20-Log – power rule

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?

`x=b^(a/3)`
`x=a^(b/3)`
`x=b^a/3`
`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 SM-Bank 13

Find `x` given `100^(x-2) = 1000^x`. (2 marks)

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`-4`

Show Worked Solution

`100^(x-2)`	`= 1000^x`
`(10^2)^(x-2)`	`= (10^3)^x`
`10^(2x-4)`	`= (10)^(3x)`
`2x-4`	`=3x`
`:. x`	`= -4`

L&E, 2ADV E1 2019 HSC 5 MC

Which of the following is equal to `(log_2 9)/(log_2 3)`?

`2`
`3`
`log_2 3`
`log_2 6`

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`A`

Show Worked Solution

`(log_2 9)/(log_2 3)`	`= (log_2 3^2)/(log_2 3)`
	`= (2 log_2 3)/(log_2 3)`
	`= 2`

`=> A`

L&E, 2ADV E1 SM-Bank 4 MC

If `f(x) = 3 log_e (2x),` and `f(5x) = log_e (y),`

then `y` is equal to

`30x`
`6x`
`125x^3`
`1000x^3`

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`D`

Show Worked Solution

`f(5x)`	`= 3 log_e (2(5x))`
`log_e (y)`	`= 3 log_e (10 x)`
	`= log_e (10x)^3`
`y`	`= 1000 x^3`

`=> D`

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)

--- 5 WORK AREA LINES (style=lined) ---

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`45 log 2`

Show Worked Solution

`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 2012 HSC 7 MC

Let `a=e^x`

Which expression is equal to `log_e(a^2)`?

`e^(2x)`
`e^(x^2)`
`2x`
`x^2`

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`C`

Show Worked Solutions

`log_e(a^2)`	`=log_e(e^x)^2`
	`=log_e(e^(2x))`
	`=2xlog_ee`
	`=2x`

`=> C`

L&E, 2ADV E1 2013 HSC 9 MC

What is the solution of `5^x=4`?

`x=(log_2 4)/5`
`x=4/(log_2 5)`
`x=(log_2 4)/(log_2 5)`
`x=log_2(4/5)`

Show Answer Only

`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`