Logarithms

Logarithm, SMB-024

Solve `log_9 27=x` for `x`. (2 marks)

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

Show Answers Only

`3/2`

Show Worked Solution

`log_9 27`	`=x`
`9^x`	`=27\ \ text{(by log definition)}`
`log_10 9^x`	`=log_10 27`
`x log_10 9`	`=log_10 27`
`x`	`=(log_10 27)/(log_10 9)`
	`=3/2`

Logarithm, SMB-023

Solve `log_16 2=x` for `x`. (2 marks)

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

Show Answers Only

`0.25`

Show Worked Solution

`log_16 2`	`=x`
`16^x`	`=2`
`log_10 16^x`	`=log_10 2`
`x log_10 16`	`=log_10 2`
`x`	`=(log_10 2)/(log_10 16)`
	`=0.25`

Logarithm, SMB-022

Solve `4^(x-1)=84` for `x`, giving your answer correct to 1 decimal place. (2 marks)

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

Show Answers Only

`4.2`

Show Worked Solution

`4^(x-1)`	`=84`
`log_10 4^(x-1)`	`=log_10 84`
`(x-1)log_10 4`	`=log_10 84`
`x-1`	`=(log_10 84)/(log_10 4)`
`x`	`=(log_10 84)/(log_10 4)+1`
	`=4.196…`
	`=4.2\ text{(to 1 d.p.)}`

Logarithm, SMB-021

Solve `3^a=28` for `a`, giving your answer correct to 2 decimal places. (2 marks)

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

Show Answers Only

`3.03`

Show Worked Solution

`3^a`	`=28`
`log_10 3^a`	`=log_10 28`
`a xx log_10 3`	`=log_10 28`
`a`	`=(log_10 28)/(log_10 3)`
	`=3.033…`
	`=3.03`

Logarithm, SMB-020

Solve `4^(x+1)=32` for `x`. (2 marks)

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

Show Answers Only

`3/2`

Show Worked Solution

`4^(x+1)`	`=32`
`log_10 4^(x+1)`	`=log_10 32`
`(x+1) log_10 4`	`=log_10 32`
`x+1`	`=(log_10 32)/(log_10 4)`
`x`	`=5/2-1`
	`=3/2`

Logarithm, SMB-019

Solve `2^t=16` . (2 marks)

Show Answers Only

`4`

Show Worked Solution

`2^t`	`=16`
`log_10 2^t`	`=log_10 16`
`t xx log_10 2`	`=log_10 16`
`t`	`=(log_10 16)/(log_10 2)`
	`=4`

Logarithm, SMB-018

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

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

Show Answers Only

`1.43`

Show Worked Solution

`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)

Show Answers Only

`0.71`

Show Worked Solution

`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)

Show Answers Only

`2.37`

Show Worked Solution

`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)

Show Answers Only

`1.642`

Show Worked Solution

`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)

Show Answers Only

`0.969`

Show Worked Solution

`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)

Show Answers Only

`1.797`

Show Worked Solution

`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`

Logarithm, SMB-012

Solve the equation `log_9 x=-3/2`. (2 marks)

Show Answers Only

`x = 1/27`

Show Worked Solution

`log_9 x`	`=-3/2`
`x`	`=9^(-3/2)`
	`=1/(sqrt9)^3`
	`=1/27`

Logarithms, SMB-011

Solve the equation `log_4 x=3/2`. (2 marks)

Show Answers Only

`x = 8`

Show Worked Solution

`log_4 x`	`=3/2`
`x`	`=4^(3/2)`
	`=8`

Logarithms, SMB-010

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

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

Show Answers Only

`x = text{−1}`

Show Worked Solution

`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-009

What is the solution to the equation `log_3(a-1) = -2`? (2 marks)

Show Answers Only

`a=10/9`

Show Worked Solution

`log_3 (a-1)`	`= -2`
`a-1`	`= 3^{-2}`
`a`	`=1/3^2+1`
	`= 10/9`

L&E, 2ADV E1 2008 HSC 7a

Solve `log_2 x-3/log_2 x=2` (3 marks)

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

Show Answers Only

`x=8\ \ text(or)\ \ 1/2`

Show Worked Solution

IMPORTANT: Students should recognise this equation as a quadratic, and the best responses substituted `log_2 x` with a variable such as `X`.

`log_2 x-3/(log_2 x)`	`=2`
`(log_2 x)^2-3`	`=2log_2 x`
`(log_2 x)^2-2log_2 x-3`	`=0`

`text(Let)\ X=log_2 x`
`:.\ X^2-2X-3`	`=0`
`(X-3)(X+1)`	`=0`

MARKER’S COMMENT: Many responses incorrectly stated that there is no solution to `log_2 x=-1` or could not find `x` given `log_2 x=3`.

`X`	`=3`	`\ \ \ \ \ \ \ \ \ \ `	`X`	`=-1`
`log_2 x`	`=3`	`\ \ \ \ \ \ \ \ \ \ `	`log_2 x`	`=-1`
`x`	`=2^3=8`	`\ \ \ \ \ \ \ \ \ \ `	`x`	`=2^{-1}=1/2`

`:.x=8\ \ text(or)\ \ 1/2`

Logarithms, SMB-008

What is the solution to the equation `log_3 x = -1`? (1 mark)

Show Answers Only

`x=1/3`

Show Worked Solution

`log_3 x`	`=-1`
`x`	`=3^{-1}`
	`=1/3`

Logarithms, SMB-007

Use the change of base formula to evaluate `log_7 13`, correct to two decimal places. (1 mark)

Show Answers Only

`1.32\ \ text{(to 2 d.p.)}`

Show Worked Solution

`log_7 13`	`= (log_10 13)/(log_10 7)`
	`= 1.3181…`
	`= 1.32\ \ text{(to 2 d.p.)}`

Logarithms, SMB-006 MC

The expression

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

is equal to

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

Show Answers Only

`B`

Show Worked Solution

`text(Solution 1)`

`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`

Logarithms, SMB-005

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

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

Show Answers Only

`4 `

Show Worked Solution

`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)

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

Show Answers Only

`10/3`

Show Worked Solution

`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)

Show Answers Only

`18/25`

Show Worked Solution

`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)

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

Show Answers Only

`x =-1/6`

Show Worked Solution

`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)`

Show Answers Only

`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`

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`

Show Answers Only

`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 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) ---

Show Answers Only

`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 2005 HSC 5a

Use the change of base formula to evaluate `log_3 7`, correct to two decimal places. (1 mark)

Show Answers Only

`1.77\ \ text{(to 2 d.p.)}`

Show Worked Solution

`log_3 7`	`= (log_10 7)/(log_10 3)`
	`= 1.771…`
	`= 1.77\ \ text{(to 2 d.p.)}`

L&E, 2ADV E1 2014 HSC 3 MC

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

`4`
`17`
`65`
`257`

Show Answers Only

`D`

Show Worked Solution

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

`=> D`

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`