Solve the equation `2 log_2(x + 5)-log_2(x + 9) = 1`. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
Aussie Maths & Science Teachers: Save your time with SmarterEd
Solve the equation `2 log_2(x + 5)-log_2(x + 9) = 1`. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`x = text{−1}`
`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})`
Solve `log_3(t)-log_3(t^2-4) = -1` for `t`. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`4 `
`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)`
Solve `e^(2 ln x) = x + 6` (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
`x = 3 or -2`
`e^(2 ln x)` | `= x + 6` |
`ln e^(2 ln x)` | `= ln (x + 6)` |
`2 ln x` | `= ln (x + 6)` |
`ln x^2` | `= ln (x + 6)` |
`x^2` | `= x + 6` |
`x^2 – x – 6` | `= 0` |
`(x – 3) (x + 2)` | `= 0` |
`:. x = 3 \ \ (x>0)`
Solve the equation `e^(4x) - 5e^(2x) + 4 = 0` for `x`
(A) `x= 1, 4`
(B) `x= – 4, – 1`
(C) `x= 0, log_e 2`
(D) `x= – log_e 2, 0, log_e 2 `
`C`
`e^(4x) – 5e^(2x) + 4 = 0`
`text(Let)\ \ X=e^(2x)`
`X^2-5X+4` | `=0` |
`(X-4)(X-1)` | `=0` |
`X` | `=4 or 1` |
`:.e^(2x)` | `=4` | `e^(2x)` | `=1` |
`2x` | `=log_e 4` | `x` | `=0` |
`x` | `=(2log_e 2)/2` | ||
`=log_e 2` |
`=> C`
Solve the following equation for `x`:
`e^(2x) + 3e^x − 10 = 0`. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
`x = ln 2`
`e^(2x) + 3e^x − 10` | `= 0` |
`:. (e^x)^2 + 3e^x − 10` | `= 0` |
`text(Let)\ \ X = e^x,`
`:. X^2 + 3X – 10` | `= 0` |
`(X + 5)(X − 2)` | `= 0` |
`:. X =2 or -5` |
`text(If)\ \ X` | `=2` |
`e^x` | `=2` |
`ln e^x` | `=ln 2` |
`x` | `=ln 2` |
`text(If)\ \ X` | `=-5` |
`e^x` | `=-5\ \ \ text{(no solution)}` |
`:. x=ln 2`
Solve the following equation for `x`:
`2e^(2x) - e^x = 0`. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
`x = ln\ 1/2`
`text(Solution 1)`
`2e^(2x) – e^x = 0`
`text(Let)\ \ X = e^x`
`2X^2 – X` | `= 0` |
`X (2X – 1)` | `= 0` |
`X = 0 or 1/2`
`text(When)\ e^x = 0\ =>\ text(no solution)`
`text(When)\ e^x = 1/2`
`ln e^x` | `= ln\ 1/2` |
`:. x` | `= ln\ 1/2` |
`text(Solution 2)`
`2e^(2x)-e^x` | `=0` |
`2e^(2x)` | `=e^x` |
`ln 2e^(2x)` | `=ln e^x` |
`ln 2 +ln e^(2x)` | `=x` |
`ln 2 + 2x` | `=x` |
`x` | `=-ln2` |
`=ln\ 1/2` |
Solve `log_e x-3/log_ex=2` (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`x=e^3\ \ text(or)\ \ e^-1`
`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` |
`X` | `=3` | `\ \ \ \ \ \ \ \ \ \ ` | `X` | `=-1` |
`log_ex` | `=3` | `\ \ \ \ \ \ \ \ \ \ ` | `log_ex` | `=-1` |
`x` | `=e^3` | `\ \ \ \ \ \ \ \ \ \ ` | `x` | `=e^-1` |
`:.x=e^3\ \ text(or)\ \ e^-1`