smc-4243-10-Product/Quotient rules

Logarithm, SMB-018

Evaluate `log_a 6` given `log_a 2=0.62` and `log_a 24=2.67`. (2 marks)

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`1.43`

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`log_a 6`	`=log_a (24/4)`
	`=log_a 24-log_b 2^2`
	`=log_a 24-2log_a 2`
	`=2.67-2 xx 0.62`
	`=1.43`

Logarithm, SMB-017

Evaluate `log_b 2` given `log_b 6=1.47` and `log_b 12=2.18`. (2 marks)

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`0.71`

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`log_b 2`	`=log_b (12/6)`
	`=log_b 12-log_b 6`
	`=2.18-1.47`
	`=0.71`

Logarithm, SMB-016

Evaluate `log_c 12` given `log_c 3=1.02` and `log_c 4=1.35`. (2 marks)

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`2.37`

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`log_c 12`	`=log_c (3xx4)`
	`=log_c 3+log_c 4`
	`=1.02+1.35`
	`=2.37`

Logarithms, SMB-015

Evaluate `log_x 20` given `log_x 2=0.458` and `log_x 5=0.726`. (2 marks)

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`1.642`

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`log_x 20`	`=log_x (4xx5)`
	`=log_x (2^2xx 5)`
	`=2log_x 2+log_x 5`
	`=2 xx 0.458 + 0.726`
	`=1.642`

Logarithms, SMB-014

Evaluate `log_a 15` given `log_a 3=0.378` and `log_a 5=0.591`. (2 marks)

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`0.969`

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`log_a 15`	`=log_a (3xx5)`
	`=log_a 3+log_a 5`
	`=0.378 + 0.591`
	`=0.969`

Logarithms, SMB-013

Evaluate `log_a 18` given `log_a 2=0.431` and `log_a 3=0.683`. (2 marks)

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`1.797`

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`log_a 18`	`=log_a (3^2xx2)`
	`=log_a 3^2+log_a 2`
	`=2log_a 3+log_a 2`
	`=2 xx 0.683 + 0.431`
	`=1.797`

Logarithms, SMB-010

Solve the equation `2 log_2(x + 5)-log_2(x + 9) = 1`. (3 marks)

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`x = text{−1}`

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`2 log_2(x + 5)-log_2(x + 9)`	`= 1`
`log_2(x + 5)^2-log_2(x + 9)`	`= 1`
`log_2(((x + 5)^2)/(x + 9))`	`= 1`
`((x + 5)^2)/(x + 9)`	`= 2`
`x^2 + 10x + 25`	`= 2x + 18`
`x^2 + 8x + 7`	`= 0`
`(x + 7)(x + 1)`	`= 0`

`:. x = -1\ \ \ \ (x != text{−7}\ \ text(as)\ \ x > text{−5})`

Logarithms, SMB-005

Solve `log_3(t)-log_3(t^2-4) = -1` for `t`. (3 marks)

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`4 `

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`log_3(t)-log_3(t^2-4)`	`= -1`
`log_3 ({t}/{t^2-4})`	`= -1`
`(t)/(t^2-4)`	`= (1)/(3)`
`t^2-4`	`= 3t`
`t^2-3t – 4`	`= 0`
`(t-4)(t+ 1)`	`= 0`

`:. t=4 \ \ \ (t > 0, \ t!= –1)`

Logarithms, SMB-004

Solve `log_2(6-x)-log_2(4-x) = 2` for `x`, where `x < 4`. (2 marks)

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`10/3`

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`text(Simplify using log laws:)`

`log_2((6-x)/(4-x))`	`= 2`
`2^2`	`= (6-x)/(4-x)`
`16-4x`	`= 6-x`
`3x`	`= 10`
`:. x`	`= 10/3`

Logarithms, SMB-003

Solve the equation `2 log_3(5)-log_3 (2) + log_3 (x) = 2` for `x.` (2 marks)

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`18/25`

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`log_3 (5)^2-log_3 (2) + log_3 (x)`	`= 2`
`log_3 (25x)-log_3 (2)`	`=2`
`log_3 ((25 x)/2)`	`= 2`
`(25x)/2`	`= 3^2`
`:. x`	`= 18/25`

Logarithms, SMB-002

Solve the equation `log_3(3x + 5) + log_3(2) = 2`, for `x`. (2 marks)

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`x =-1/6`

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`text(Simplify using log laws:)`

`log_3(2(3x + 5))`	`=2`
`log_3(6x + 10)`	`=2`
`6x +10`	`=9`
`6x`	`= -1`
`x`	`=-1/6`

Logarithms, SMB-001 MC

It is given that `log_10 a = log_10 b-log_10 c`, where `a, b, c > 0.`

Which statement is true?

`a = b-c`
`a = b/c`
`log_10 a = b/c`
`log_10 a = (log_10 b)/(log_10 c)`

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`B`

Show Worked Solution

Mean mark 51%.
COMMENT: Use of log laws here proved difficult for many students.

`log_10 a`	`= log_10 b-log_10 c`
`log_10 a`	`= log_10 (b/c)`
`:. a`	`= b/c`

`=> B`