What is the general solution for 75=50cosh(x/(50)) ?

The general solution for 75=50cosh(x/(50)) is x=50ln((3+sqrt(5))/2)

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75=50cosh(x/(50))

Solution

75 = 50 cosh (\frac{x}{50})

Solution

x = 50 ln (\frac{3 + 5}{2})

Degrees

x = 2757.14066 \dots^{\circ}

Solution steps

75 = 50 cosh (\frac{x}{50})

Switch sides

50 cosh (\frac{x}{50}) = 75

Rewrite using trig identities

50 cosh (\frac{x}{50}) = 75

Use the Hyperbolic identity:

cosh (x) = \frac{e ^{x} + e ^{- x}}{2}

50 \cdot \frac{e ^{\frac{x}{50}} + e ^{- \frac{x}{50}}}{2} = 75

50 \cdot \frac{e ^{\frac{x}{50}} + e ^{- \frac{x}{50}}}{2} = 75

50 \cdot \frac{e ^{\frac{x}{50}} + e ^{- \frac{x}{50}}}{2} = 75 : x = 50 ln (\frac{3 + 5}{2})

50 \cdot \frac{e ^{\frac{x}{50}} + e ^{- \frac{x}{50}}}{2} = 75

Apply exponent rules

50 \cdot \frac{e ^{\frac{x}{50}} + e ^{- \frac{x}{50}}}{2} = 75

Apply exponent rule:

a^{b c} = (a^{b})^{c}

e^{\frac{x}{50}} = (e^{x})^{0.02}, e^{- \frac{x}{50}} = (e^{x})^{- 0.02}

50 \cdot \frac{( e ^{x} ) ^{0.02} + ( e ^{x} ) ^{- 0.02}}{2} = 75

50 \cdot \frac{( e ^{x} ) ^{0.02} + ( e ^{x} ) ^{- 0.02}}{2} = 75

Rewrite the equation with

e^{x} = u

50 \cdot \frac{( u ) ^{0.02} + ( u ) ^{- 0.02}}{2} = 75

Solve

50 \cdot \frac{u ^{0.02} + u ^{- 0.02}}{2} = 75 : u = \frac{( 3 + 5 ) ^{50}}{2 ^{50}}, u = \frac{( 3 - 5 ) ^{50}}{2 ^{50}}

50 \cdot \frac{u ^{0.02} + u ^{- 0.02}}{2} = 75

Expand

50 \cdot \frac{u ^{0.02} + u ^{- 0.02}}{2} : 25 u^{0.02} + \frac{25}{u ^{0.02}}

50 \cdot \frac{u ^{0.02} + u ^{- 0.02}}{2}

\frac{u ^{0.02} + u ^{- 0.02}}{2} = \frac{u ^{0.04} + 1}{2 u ^{0.02}}

\frac{u ^{0.02} + u ^{- 0.02}}{2}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= \frac{u ^{0.02} + \frac{1}{u ^{0.02}}}{2}

Join

u^{0.02} + \frac{1}{u ^{0.02}} : \frac{u ^{0.04} + 1}{u ^{0.02}}

u^{0.02} + \frac{1}{u ^{0.02}}

Convert element to fraction:

u^{0.02} = \frac{u ^{0.02} u ^{0.02}}{u ^{0.02}}

= \frac{u ^{0.02} u ^{0.02}}{u ^{0.02}} + \frac{1}{u ^{0.02}}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{u ^{0.02} u ^{0.02} + 1}{u ^{0.02}}

u^{0.02} u^{0.02} + 1 = u^{0.04} + 1

u^{0.02} u^{0.02} + 1

u^{0.02} u^{0.02} = u^{0.04}

u^{0.02} u^{0.02}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{0.02} u^{0.02} = u^{0.02 + 0.02}

= u^{0.02 + 0.02}

Add the numbers:

0.02 + 0.02 = 0.04

= u^{0.04}

= u^{0.04} + 1

= \frac{u ^{0.04} + 1}{u ^{0.02}}

= \frac{\frac{u ^{0.04} + 1}{u ^{0.02}}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{u ^{0.04} + 1}{u ^{0.02} \cdot 2}

= 50 \cdot \frac{u ^{0.04} + 1}{2 u ^{0.02}}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( u ^{0.04} + 1 ) \cdot 50}{u ^{0.02} \cdot 2}

Divide the numbers:

\frac{50}{2} = 25

= \frac{25 ( u ^{0.04} + 1 )}{u ^{0.02}}

Expand

25 (u^{0.04} + 1) : 25 u^{0.04} + 25

25 (u^{0.04} + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = 25, b = u^{0.04}, c = 1

= 25 u^{0.04} + 25 \cdot 1

Multiply the numbers:

25 \cdot 1 = 25

= 25 u^{0.04} + 25

= \frac{25 u ^{0.04} + 25}{u ^{0.02}}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{25 u ^{0.04} + 25}{u ^{0.02}} = \frac{25 u ^{0.04}}{u ^{0.02}} + \frac{25}{u ^{0.02}}

= \frac{25 u ^{0.04}}{u ^{0.02}} + \frac{25}{u ^{0.02}}

Cancel

\frac{25 u ^{0.04}}{u ^{0.02}} : 25 u^{0.02}

\frac{25 u ^{0.04}}{u ^{0.02}}

Cancel

\frac{25 u ^{0.04}}{u ^{0.02}} : 25 u^{0.02}

\frac{25 u ^{0.04}}{u ^{0.02}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{u ^{0.04}}{u ^{0.02}} = u^{0.04 - 0.02}

= 25 u^{0.04 - 0.02}

Subtract the numbers:

0.04 - 0.02 = 0.02

= 25 u^{0.02}

= 25 u^{0.02}

= 25 u^{0.02} + \frac{25}{u ^{0.02}}

25 u^{0.02} + \frac{25}{u ^{0.02}} = 75

Rewrite the equation with

u^{\frac{1}{50}} = v

25 v + \frac{25}{v} = 75

Solve

25 v + \frac{25}{v} = 75 : v = \frac{3 + 5}{2}, v = \frac{3 - 5}{2}

25 v + \frac{25}{v} = 75

Multiply both sides by

v

25 v + \frac{25}{v} = 75

Multiply both sides by

v

25 vv + \frac{25}{v} v = 75 v

Simplify

25 vv + \frac{25}{v} v = 75 v

Simplify

25 vv : 25 v^{2}

25 vv

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

vv = v^{1 + 1}

= 25 v^{1 + 1}

Add the numbers:

1 + 1 = 2

= 25 v^{2}

Simplify

\frac{25}{v} v : 25

\frac{25}{v} v

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{25 v}{v}

Cancel the common factor:

v

= 25

25 v^{2} + 25 = 75 v

25 v^{2} + 25 = 75 v

25 v^{2} + 25 = 75 v

Solve

25 v^{2} + 25 = 75 v : v = \frac{3 + 5}{2}, v = \frac{3 - 5}{2}

25 v^{2} + 25 = 75 v

Move

75 v

to the left side

25 v^{2} + 25 = 75 v

Subtract

75 v

from both sides

25 v^{2} + 25 - 75 v = 75 v - 75 v

Simplify

25 v^{2} + 25 - 75 v = 0

25 v^{2} + 25 - 75 v = 0

Write in the standard form

a x^{2} + b x + c = 0

25 v^{2} - 75 v + 25 = 0

Solve with the quadratic formula

25 v^{2} - 75 v + 25 = 0

Quadratic Equation Formula:

For

a = 25, b = - 75, c = 25

v_{1, 2} = \frac{- ( - 75 ) \pm ( - 75 ) ^{2} - 4 \cdot 25 \cdot 25}{2 \cdot 25}

v_{1, 2} = \frac{- ( - 75 ) \pm ( - 75 ) ^{2} - 4 \cdot 25 \cdot 25}{2 \cdot 25}

(- 75)^{2} - 4 \cdot 25 \cdot 25 = 255

(- 75)^{2} - 4 \cdot 25 \cdot 25

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 75)^{2} = 7 5^{2}

= 7 5^{2} - 4 \cdot 25 \cdot 25

Multiply the numbers:

4 \cdot 25 \cdot 25 = 2500

= 7 5^{2} - 2500

7 5^{2} = 5625

= 5625 - 2500

Subtract the numbers:

5625 - 2500 = 3125

= 3125

Prime factorization of

3125 : 5^{5}

3125

3125

divides by

53125 = 625 \cdot 5

= 5 \cdot 625

625

divides by

5625 = 125 \cdot 5

= 5 \cdot 5 \cdot 125

125

divides by

5125 = 25 \cdot 5

= 5 \cdot 5 \cdot 5 \cdot 25

25

divides by

525 = 5 \cdot 5

= 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5

5

is a prime number, therefore no further factorization is possible

= 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5

= 5^{5}

= 5^{5}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 5^{4} \cdot 5

Apply radical rule:

= 5 5^{4}

Apply radical rule:

5^{4} = 5^{\frac{4}{2}} = 5^{2}

= 5^{2} 5

Refine

= 255

v_{1, 2} = \frac{- ( - 75 ) \pm 25 5}{2 \cdot 25}

Separate the solutions

v_{1} = \frac{- ( - 75 ) + 25 5}{2 \cdot 25}, v_{2} = \frac{- ( - 75 ) - 25 5}{2 \cdot 25}

v = \frac{- ( - 75 ) + 25 5}{2 \cdot 25} : \frac{3 + 5}{2}

\frac{- ( - 75 ) + 25 5}{2 \cdot 25}

Apply rule

- (- a) = a

= \frac{75 + 25 5}{2 \cdot 25}

Multiply the numbers:

2 \cdot 25 = 50

= \frac{75 + 25 5}{50}

Factor

75 + 255 : 25 (3 + 5)

75 + 255

Rewrite as

= 25 \cdot 3 + 255

Factor out common term

25

= 25 (3 + 5)

= \frac{25 ( 3 + 5 )}{50}

Cancel the common factor:

25

= \frac{3 + 5}{2}

v = \frac{- ( - 75 ) - 25 5}{2 \cdot 25} : \frac{3 - 5}{2}

\frac{- ( - 75 ) - 25 5}{2 \cdot 25}

Apply rule

- (- a) = a

= \frac{75 - 25 5}{2 \cdot 25}

Multiply the numbers:

2 \cdot 25 = 50

= \frac{75 - 25 5}{50}

Factor

75 - 255 : 25 (3 - 5)

75 - 255

Rewrite as

= 25 \cdot 3 - 255

Factor out common term

25

= 25 (3 - 5)

= \frac{25 ( 3 - 5 )}{50}

Cancel the common factor:

25

= \frac{3 - 5}{2}

The solutions to the quadratic equation are:

v = \frac{3 + 5}{2}, v = \frac{3 - 5}{2}

v = \frac{3 + 5}{2}, v = \frac{3 - 5}{2}

Verify Solutions

Find undefined (singularity) points:

v = 0

Take the denominator(s) of

25 v + \frac{25}{v}

and compare to zero

v = 0

The following points are undefined

v = 0

Combine undefined points with solutions:

v = \frac{3 + 5}{2}, v = \frac{3 - 5}{2}

v = \frac{3 + 5}{2}, v = \frac{3 - 5}{2}

Substitute back

v = u^{\frac{1}{50}},

solve for

u

Solve

u^{\frac{1}{50}} = \frac{3 + 5}{2} : u = \frac{( 3 + 5 ) ^{50}}{2 ^{50}}

u^{\frac{1}{50}} = \frac{3 + 5}{2}

Take both sides of the equation to the power of

50 : u = \frac{( 3 + 5 ) ^{50}}{2 ^{50}}

u^{\frac{1}{50}} = \frac{3 + 5}{2}

(u^{\frac{1}{50}})^{50} = (\frac{3 + 5}{2})^{50}

Expand

(u^{\frac{1}{50}})^{50} : u

(u^{\frac{1}{50}})^{50}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= u^{\frac{1}{50} \cdot 50}

\frac{1}{50} \cdot 50 = 1

\frac{1}{50} \cdot 50

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 50}{50}

Cancel the common factor:

50

= 1

= u

Expand

(\frac{3 + 5}{2})^{50} : \frac{( 3 + 5 ) ^{50}}{2 ^{50}}

(\frac{3 + 5}{2})^{50}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 + 5 ) ^{50}}{2 ^{50}}

u = \frac{( 3 + 5 ) ^{50}}{2 ^{50}}

u = \frac{( 3 + 5 ) ^{50}}{2 ^{50}}

Verify Solutions:

u = \frac{( 3 + 5 ) ^{50}}{2 ^{50}}

True

Check the solutions by plugging them into

u^{\frac{1}{50}} = \frac{3 + 5}{2}

Remove the ones that don't agree with the equation.

Plug in

u = \frac{( 3 + 5 ) ^{50}}{2 ^{50}} :

True

(\frac{( 3 + 5 ) ^{50}}{2 ^{50}})^{\frac{1}{50}} = \frac{3 + 5}{2}

(\frac{( 3 + 5 ) ^{50}}{2 ^{50}})^{\frac{1}{50}} = \frac{3 + 5}{2}

(\frac{( 3 + 5 ) ^{50}}{2 ^{50}})^{\frac{1}{50}}

Apply radical rule:

assuming

a \geq 0, b \geq 0

Apply radical rule:

assuming

a \geq 0

Apply radical rule:

assuming

a \geq 0

= \frac{3 + 5}{2}

\frac{3 + 5}{2} = \frac{3 + 5}{2}

True

The solution is

u = \frac{( 3 + 5 ) ^{50}}{2 ^{50}}

Solve

u^{\frac{1}{50}} = \frac{3 - 5}{2} : u = \frac{( 3 - 5 ) ^{50}}{2 ^{50}}

u^{\frac{1}{50}} = \frac{3 - 5}{2}

Take both sides of the equation to the power of

50 : u = \frac{( 3 - 5 ) ^{50}}{2 ^{50}}

u^{\frac{1}{50}} = \frac{3 - 5}{2}

(u^{\frac{1}{50}})^{50} = (\frac{3 - 5}{2})^{50}

Expand

(u^{\frac{1}{50}})^{50} : u

(u^{\frac{1}{50}})^{50}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= u^{\frac{1}{50} \cdot 50}

\frac{1}{50} \cdot 50 = 1

\frac{1}{50} \cdot 50

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 50}{50}

Cancel the common factor:

50

= 1

= u

Expand

(\frac{3 - 5}{2})^{50} : \frac{( 3 - 5 ) ^{50}}{2 ^{50}}

(\frac{3 - 5}{2})^{50}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 - 5 ) ^{50}}{2 ^{50}}

u = \frac{( 3 - 5 ) ^{50}}{2 ^{50}}

u = \frac{( 3 - 5 ) ^{50}}{2 ^{50}}

Verify Solutions:

u = \frac{( 3 - 5 ) ^{50}}{2 ^{50}}

True

Check the solutions by plugging them into

u^{\frac{1}{50}} = \frac{3 - 5}{2}

Remove the ones that don't agree with the equation.

Plug in

u = \frac{( 3 - 5 ) ^{50}}{2 ^{50}} :

True

(\frac{( 3 - 5 ) ^{50}}{2 ^{50}})^{\frac{1}{50}} = \frac{3 - 5}{2}

(\frac{( 3 - 5 ) ^{50}}{2 ^{50}})^{\frac{1}{50}} = \frac{3 - 5}{2}

(\frac{( 3 - 5 ) ^{50}}{2 ^{50}})^{\frac{1}{50}}

Apply radical rule:

assuming

a \geq 0, b \geq 0

Apply radical rule:

assuming

a \geq 0

Apply radical rule:

assuming

a \geq 0

= \frac{3 - 5}{2}

\frac{3 - 5}{2} = \frac{3 - 5}{2}

True

The solution is

u = \frac{( 3 - 5 ) ^{50}}{2 ^{50}}

u = \frac{( 3 + 5 ) ^{50}}{2 ^{50}}, u = \frac{( 3 - 5 ) ^{50}}{2 ^{50}}

Verify Solutions:

u = \frac{( 3 + 5 ) ^{50}}{2 ^{50}}

True

, u = \frac{( 3 - 5 ) ^{50}}{2 ^{50}}

True

Check the solutions by plugging them into

50 \frac{u ^{0.02} + u ^{- 0.02}}{2} = 75

Remove the ones that don't agree with the equation.

Plug in

u = \frac{( 3 + 5 ) ^{50}}{2 ^{50}} :

True

50 \cdot \frac{( \frac{( 3 + 5 ) ^{50}}{2 ^{50}} ) ^{0.02} + ( \frac{( 3 + 5 ) ^{50}}{2 ^{50}} ) ^{- 0.02}}{2} = 75

50 \cdot \frac{( \frac{( 3 + 5 ) ^{50}}{2 ^{50}} ) ^{0.02} + ( \frac{( 3 + 5 ) ^{50}}{2 ^{50}} ) ^{- 0.02}}{2} = 75

50 \cdot \frac{( \frac{( 3 + 5 ) ^{50}}{2 ^{50}} ) ^{0.02} + ( \frac{( 3 + 5 ) ^{50}}{2 ^{50}} ) ^{- 0.02}}{2}

\frac{( \frac{( 3 + 5 ) ^{50}}{2 ^{50}} ) ^{0.02} + ( \frac{( 3 + 5 ) ^{50}}{2 ^{50}} ) ^{- 0.02}}{2} = \frac{3}{2}

\frac{( \frac{( 3 + 5 ) ^{50}}{2 ^{50}} ) ^{0.02} + ( \frac{( 3 + 5 ) ^{50}}{2 ^{50}} ) ^{- 0.02}}{2}

(\frac{( 3 + 5 ) ^{50}}{2 ^{50}})^{0.02} = 2.61803 \dots

(\frac{( 3 + 5 ) ^{50}}{2 ^{50}})^{0.02}

\frac{( 3 + 5 ) ^{50}}{2 ^{50}} = 7.92071 E 20

\frac{( 3 + 5 ) ^{50}}{2 ^{50}}

Convert element to a decimal form

2^{50} = 1.1259 E 15

= \frac{( 3 + 5 ) ^{50}}{1.1259 E 15}

Convert element to a decimal form

(3 + 5)^{50} = 8.91792 E 35

= \frac{8.91792 E 35}{1.1259 E 15}

Divide the numbers:

\frac{8.91792 E 35}{1.1259 E 15} = 7.92071 E 20

= 7.92071 E 20

= 7.92071 E 2 0^{0.02}

7.92071 E 2 0^{0.02} = 2.61803 \dots

= 2.61803 \dots

(\frac{( 3 + 5 ) ^{50}}{2 ^{50}})^{- 0.02} = 0.38196 \dots

(\frac{( 3 + 5 ) ^{50}}{2 ^{50}})^{- 0.02}

\frac{( 3 + 5 ) ^{50}}{2 ^{50}} = 7.92071 E 20

\frac{( 3 + 5 ) ^{50}}{2 ^{50}}

Convert element to a decimal form

2^{50} = 1.1259 E 15

= \frac{( 3 + 5 ) ^{50}}{1.1259 E 15}

Convert element to a decimal form

(3 + 5)^{50} = 8.91792 E 35

= \frac{8.91792 E 35}{1.1259 E 15}

Divide the numbers:

\frac{8.91792 E 35}{1.1259 E 15} = 7.92071 E 20

= 7.92071 E 20

= 7.92071 E 2 0^{- 0.02}

7.92071 E 2 0^{- 0.02} = 0.38196 \dots

= 0.38196 \dots

= \frac{2.61803 \dots + 0.38196 \dots}{2}

Add the numbers:

2.61803 \dots + 0.38196 \dots = 3

= \frac{3}{2}

= 50 \cdot \frac{3}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 \cdot 50}{2}

Multiply the numbers:

3 \cdot 50 = 150

= \frac{150}{2}

Divide the numbers:

\frac{150}{2} = 75

= 75

75 = 75

True

Plug in

u = \frac{( 3 - 5 ) ^{50}}{2 ^{50}} :

True

50 \cdot \frac{( \frac{( 3 - 5 ) ^{50}}{2 ^{50}} ) ^{0.02} + ( \frac{( 3 - 5 ) ^{50}}{2 ^{50}} ) ^{- 0.02}}{2} = 75

50 \cdot \frac{( \frac{( 3 - 5 ) ^{50}}{2 ^{50}} ) ^{0.02} + ( \frac{( 3 - 5 ) ^{50}}{2 ^{50}} ) ^{- 0.02}}{2} = 75

50 \cdot \frac{( \frac{( 3 - 5 ) ^{50}}{2 ^{50}} ) ^{0.02} + ( \frac{( 3 - 5 ) ^{50}}{2 ^{50}} ) ^{- 0.02}}{2}

\frac{( \frac{( 3 - 5 ) ^{50}}{2 ^{50}} ) ^{0.02} + ( \frac{( 3 - 5 ) ^{50}}{2 ^{50}} ) ^{- 0.02}}{2} = \frac{3}{2}

\frac{( \frac{( 3 - 5 ) ^{50}}{2 ^{50}} ) ^{0.02} + ( \frac{( 3 - 5 ) ^{50}}{2 ^{50}} ) ^{- 0.02}}{2}

(\frac{( 3 - 5 ) ^{50}}{2 ^{50}})^{0.02} = 0.38196 \dots

(\frac{( 3 - 5 ) ^{50}}{2 ^{50}})^{0.02}

\frac{( 3 - 5 ) ^{50}}{2 ^{50}} = 1.26251 E - 21

\frac{( 3 - 5 ) ^{50}}{2 ^{50}}

Convert element to a decimal form

2^{50} = 1.1259 E 15

= \frac{( 3 - 5 ) ^{50}}{1.1259 E 15}

Convert element to a decimal form

(3 - 5)^{50} = 1.42146 E - 6

= \frac{1.42146 E - 6}{1.1259 E 15}

Divide the numbers:

\frac{1.42146 E - 6}{1.1259 E 15} = 1.26251 E - 21

= 1.26251 E - 21

= 1.26251 E - 2 1^{0.02}

1.26251 E - 2 1^{0.02} = 0.38196 \dots

= 0.38196 \dots

(\frac{( 3 - 5 ) ^{50}}{2 ^{50}})^{- 0.02} = 2.61803 \dots

(\frac{( 3 - 5 ) ^{50}}{2 ^{50}})^{- 0.02}

\frac{( 3 - 5 ) ^{50}}{2 ^{50}} = 1.26251 E - 21

\frac{( 3 - 5 ) ^{50}}{2 ^{50}}

Convert element to a decimal form

2^{50} = 1.1259 E 15

= \frac{( 3 - 5 ) ^{50}}{1.1259 E 15}

Convert element to a decimal form

(3 - 5)^{50} = 1.42146 E - 6

= \frac{1.42146 E - 6}{1.1259 E 15}

Divide the numbers:

\frac{1.42146 E - 6}{1.1259 E 15} = 1.26251 E - 21

= 1.26251 E - 21

= 1.26251 E - 2 1^{- 0.02}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{1.26251 E - 2 1 ^{0.02}}

1.26251 E - 2 1^{0.02} = 0.38196 \dots

= \frac{1}{0.38196 \dots}

Divide the numbers:

\frac{1}{0.38196 \dots} = 2.61803 \dots

= 2.61803 \dots

= \frac{0.38196 \dots + 2.61803 \dots}{2}

Add the numbers:

0.38196 \dots + 2.61803 \dots = 3

= \frac{3}{2}

= 50 \cdot \frac{3}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 \cdot 50}{2}

Multiply the numbers:

3 \cdot 50 = 150

= \frac{150}{2}

Divide the numbers:

\frac{150}{2} = 75

= 75

75 = 75

True

The solutions are

u = \frac{( 3 + 5 ) ^{50}}{2 ^{50}}, u = \frac{( 3 - 5 ) ^{50}}{2 ^{50}}

u = \frac{( 3 + 5 ) ^{50}}{2 ^{50}}, u = \frac{( 3 - 5 ) ^{50}}{2 ^{50}}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = \frac{( 3 + 5 ) ^{50}}{2 ^{50}} : x = 50 ln (\frac{3 + 5}{2})

e^{x} = \frac{( 3 + 5 ) ^{50}}{2 ^{50}}

Apply exponent rules

e^{x} = \frac{( 3 + 5 ) ^{50}}{2 ^{50}}

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{2 ^{50}} = 2^{- 50}

e^{x} = (3 + 5)^{50} \cdot 2^{- 50}

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln ((3 + 5)^{50} \cdot 2^{- 50})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln ((3 + 5)^{50} \cdot 2^{- 50})

Simplify

ln ((3 + 5)^{50} \cdot 2^{- 50}) : 50 ln (\frac{3 + 5}{2})

ln ((3 + 5)^{50} \cdot 2^{- 50})

Multiply

(3 + 5)^{50} \cdot 2^{- 50} : \frac{( 3 + 5 ) ^{50}}{2 ^{50}}

(3 + 5)^{50} \cdot 2^{- 50}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

2^{- 50} = \frac{1}{2 ^{50}}

= \frac{1}{2 ^{50}} (3 + 5)^{50}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot ( 3 + 5 ) ^{50}}{2 ^{50}}

Multiply:

1 \cdot (3 + 5)^{50} = (3 + 5)^{50}

= \frac{( 3 + 5 ) ^{50}}{2 ^{50}}

= ln (\frac{( 3 + 5 ) ^{50}}{2 ^{50}})

Combine same powers :

\frac{x ^{n}}{y ^{n}} = (\frac{x}{y})^{n}

= ln (\frac{3 + 5}{2})^{50}

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

= 50 ln (\frac{3 + 5}{2})

x = 50 ln (\frac{3 + 5}{2})

x = 50 ln (\frac{3 + 5}{2})

Solve

e^{x} = \frac{( 3 - 5 ) ^{50}}{2 ^{50}} :

No Solution for

x \in R

e^{x} = \frac{( 3 - 5 ) ^{50}}{2 ^{50}}

Apply exponent rules

e^{x} = \frac{( 3 - 5 ) ^{50}}{2 ^{50}}

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{2 ^{50}} = 2^{- 50}

e^{x} = (3 - 5)^{50} \cdot 2^{- 50}

e^{x} = (3 - 5)^{50} \cdot 2^{- 50}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = 50 ln (\frac{3 + 5}{2})

x = 50 ln (\frac{3 + 5}{2})

Popular Examples

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Frequently Asked Questions (FAQ)

What is the general solution for 75=50cosh(x/(50)) ?
The general solution for 75=50cosh(x/(50)) is x=50ln((3+sqrt(5))/2)