- Solve the equation (1/2)
^{2x + 1}= 1

- Solve x y
^{m}= y x^{3}for m.

- Given: log
_{8}(5) = b. Express log_{4}(10) in terms of b.

- Simplify without calculator: log
_{6}(216) + [ log(42) – log(6) ] / log(49)

- Simplify without calculator: ((3
^{-1}– 9^{-1}) / 6)^{1/3}

- Express (log
_{x}a)(log_{a}b) as a single logarithm.

- Find
**a**so that the graph of y = logx passes through the point (e , 2)._{a}

- Find constant
**A**such that log_{3}x =**A**log_{5}x

for all x > 0.

- Solve for x the equation log [ log (2 + log
_{2}(x + 1)) ] = 0

- Solve for x the equation 2 x b
^{4 log}_{b}^{x}= 486

- Solve for x the equation ln (x – 1) + ln (2x – 1) = 2 ln (x + 1)

- Find the x intercept of the graph of y = 2 log( sqrt(x – 1) – 2)

- Solve for x the equation 9
^{x}– 3^{x}– 8 = 0

- Solve for x the equation 4
^{x – 2}= 3^{x + 4}

15. If log_{x}(1 / 8) = -3 / 4, than what is x?

Solutions to the Above Problems

^{2x + 1}= (1/2)

^{0}

Leads to 2x + 1 = 0

Solve for x: x = -1/2

^{m – 1}= x

^{2}

Take ln of both sides (m – 1) ln y = 2 ln x

Solve for m: m = 1 + 2 ln(x) / ln(y)

_{4}(10) = log

_{4}(2) + log

_{4}(5)

log

_{4}(2) = log

_{4}(4

^{1/2}) = 1/2

Use change of base formula to write: log

_{4}(5) = log

_{8}(5) / log

_{8}(4) = b / (2/3) , since log

_{8}(4) = 2/3

log

_{4}(10) = log

_{4}(2) + log

_{4}(5) = (1 + 3b) / 2

_{6}(216) + [ log(42) – log(6) ] / log(49)

= log

_{6}(6

^{3}) + log(42/6) / log(7

^{2})

= 3 + log(7) /2 log(7) = 3 + 1/2 = 7/2

^{-1}– 9

^{-1}) / 6)

^{1/3}

= ((1/3 – 1/9) / 6)

^{1/3}

= ((6 / 27) / 6)

^{1/3}= 1/3

_{x}a)(log

_{a}b)

= log

_{x}a (log

_{x}b / log

_{x}a) = log

_{x}b

_{a}e

a

^{2}= e

ln(a

^{2}) = ln e

2 ln a = 1

a = e

^{1/2}

ln (x) / ln(3) = A ln(x) / ln(5)

A = ln(5) / ln(3)

_{2}(x + 1)) ] = log (1) , since log(1) = 0.

log (2 + log

_{2}(x + 1)) = 1

2 + log

_{2}(x + 1) = 10

log

_{2}(x + 1) = 8

x + 1 = 2

^{8}

x = 2

^{8}– 1

^{4 logbx}= x

^{4}

The given equation may be written as: 2x x

^{4}= 486

x = 243

^{1/5}= 3

^{2}

ln function is a one to one function, hence: (x – 1)(2x – 1) = (x + 1)

^{2}

Solve the above quadratic function: x = 0 and x = 5

Only x = 5 is a valid solution to the equation given above since x = 0 is not in the domain of the expressions making the equations.

Divide both sides by 2: log( sqrt(x – 1) – 2) = 0

Use the fact that log(1)= 0: sqrt(x – 1) – 2 = 1

Rewrite as: sqrt(x – 1) = 3

Raise both sides to the power 2: (x – 1) = 3

^{2}

x – 1 = 9

x = 10

^{x}– 3

^{x}– 8 = 0

Note that: 9

^{x}= (3

^{x})

^{2}

Equation may be written as: (3

^{x})

^{2}– 3

^{x}– 8 = 0

Let y = 3

^{x}and rewite equation with y: y

^{2}– y – 8 = 0

Solve for y: y = ( 1 + sqrt(33) ) / 2 and ( 1 – sqrt(33) ) / 2

Since y = 3

^{x}, the only acceptable solution is y = ( 1 + sqrt(33) ) / 2

3

^{x}= ( 1 + sqrt(33) ) / 2

Use ln on both sides: ln 3

^{x}= ln [ ( 1 + sqrt(33) ) / 2] Simplify and solve: x = ln [ ( 1 + sqrt(33) ) / 2] / ln 3

^{x – 2}= 3

^{x + 4}

Take ln of both sides: ln ( 4

^{x – 2}) = ln ( 3

^{x + 4})

Simplify: (x – 2) ln 4 = (x + 4) ln 3

Expand: x ln 4 – 2 ln 4 = x ln 3 + 4 ln 3

Group like terms: x ln 4 – x ln 3 = 4 ln 3 + 2 ln 4

Solve for x: x = ( 4 ln 3 + 2 ln 4 ) / (ln 4 – ln 3) = ln (3

^{4}* 4

^{2}) / ln (4/3) = ln (3

^{4}* 2

^{4}) / ln (4/3)

= 4 ln(6) / ln(4/3)

^{– 3 / 4}= 1 / 8

Raise both sides of the above equation to the power -4 / 3: (x

^{– 3 / 4})

^{– 4 / 3}= (1 / 8)

^{– 4 / 3}

simplify: x = 8

^{4 / 3}= 2

^{4}= 16