What is the general solution for 5^{sin(2x-x/4)}=1 ?

The general solution for 5^{sin(2x-x/4)}=1 is x=(8pin)/7 ,x=(4pi)/7+(8pin)/7

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5^{sin(2x-x/4)}=1

Solution

5^{s i n (2 x - \frac{x}{4})} = 1

Solution

x = \frac{8 πn}{7}, x = \frac{4 π}{7} + \frac{8 πn}{7}

Degrees

x = 0^{\circ} + 205.71428 \dots^{\circ} n, x = 102.85714 \dots^{\circ} + 205.71428 \dots^{\circ} n

Solution steps

5^{s i n (2 x - \frac{x}{4})} = 1

Apply exponent rules

5^{s i n (2 x - \frac{x}{4})} = 1

f (x) = g (x)

, then

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

ln (5^{s i n (2 x - \frac{x}{4})}) = ln (1)

Apply log rule:

ln (x^{a}) = a \cdot ln (x)

ln (5^{s i n (2 x - \frac{x}{4})}) = sin (2 x - \frac{x}{4}) ln (5)

sin (2 x - \frac{x}{4}) ln (5) = ln (1)

sin (2 x - \frac{x}{4}) ln (5) = ln (1)

Solve

sin (2 x - \frac{x}{4}) ln (5) = ln (1) : sin (2 x - \frac{x}{4}) = 0

sin (2 x - \frac{x}{4}) ln (5) = ln (1)

Divide both sides by

ln (5)

sin (2 x - \frac{x}{4}) ln (5) = ln (1)

Divide both sides by

ln (5)

\frac{sin ( 2 x - \frac{x}{4} ) ln ( 5 )}{ln ( 5 )} = \frac{ln ( 1 )}{ln ( 5 )}

Simplify

\frac{sin ( 2 x - \frac{x}{4} ) ln ( 5 )}{ln ( 5 )} = \frac{ln ( 1 )}{ln ( 5 )}

Simplify

\frac{sin ( 2 x - \frac{x}{4} ) ln ( 5 )}{ln ( 5 )} : sin (2 x - \frac{x}{4})

\frac{sin ( 2 x - \frac{x}{4} ) ln ( 5 )}{ln ( 5 )}

Cancel the common factor:

ln (5)

= sin (2 x - \frac{x}{4})

Simplify

\frac{ln ( 1 )}{ln ( 5 )} : 0

\frac{ln ( 1 )}{ln ( 5 )}

Simplify

ln (1) : 0

ln (1)

Apply log rule:

lo g_{a} (1) = 0

= 0

= \frac{0}{ln ( 5 )}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

sin (2 x - \frac{x}{4}) = 0

sin (2 x - \frac{x}{4}) = 0

sin (2 x - \frac{x}{4}) = 0

sin (2 x - \frac{x}{4}) = 0

General solutions for

sin (2 x - \frac{x}{4}) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

2 x - \frac{x}{4} = 0 + 2 πn, 2 x - \frac{x}{4} = π + 2 πn

2 x - \frac{x}{4} = 0 + 2 πn, 2 x - \frac{x}{4} = π + 2 πn

Solve

2 x - \frac{x}{4} = 0 + 2 πn : x = \frac{8 πn}{7}

2 x - \frac{x}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

2 x - \frac{x}{4} = 2 πn

Multiply both sides by

4

2 x - \frac{x}{4} = 2 πn

Multiply both sides by

4

2 x \cdot 4 - \frac{x}{4} \cdot 4 = 2 πn \cdot 4

Simplify

2 x \cdot 4 - \frac{x}{4} \cdot 4 = 2 πn \cdot 4

Simplify

2 x \cdot 4 : 8 x

2 x \cdot 4

Multiply the numbers:

2 \cdot 4 = 8

= 8 x

Simplify

- \frac{x}{4} \cdot 4 : - x

- \frac{x}{4} \cdot 4

Multiply fractions:

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

= - \frac{x \cdot 4}{4}

Cancel the common factor:

4

= - x

Simplify

2 πn \cdot 4 : 8 πn

2 πn \cdot 4

Multiply the numbers:

2 \cdot 4 = 8

= 8 πn

8 x - x = 8 πn

7 x = 8 πn

7 x = 8 πn

7 x = 8 πn

Divide both sides by

7

7 x = 8 πn

Divide both sides by

7

\frac{7 x}{7} = \frac{8 πn}{7}

Simplify

x = \frac{8 πn}{7}

x = \frac{8 πn}{7}

Solve

2 x - \frac{x}{4} = π + 2 πn : x = \frac{4 π}{7} + \frac{8 πn}{7}

2 x - \frac{x}{4} = π + 2 πn

Multiply both sides by

4

2 x - \frac{x}{4} = π + 2 πn

Multiply both sides by

4

2 x \cdot 4 - \frac{x}{4} \cdot 4 = π 4 + 2 πn \cdot 4

Simplify

2 x \cdot 4 - \frac{x}{4} \cdot 4 = π 4 + 2 πn \cdot 4

Simplify

2 x \cdot 4 : 8 x

2 x \cdot 4

Multiply the numbers:

2 \cdot 4 = 8

= 8 x

Simplify

- \frac{x}{4} \cdot 4 : - x

- \frac{x}{4} \cdot 4

Multiply fractions:

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

= - \frac{x \cdot 4}{4}

Cancel the common factor:

4

= - x

Simplify

π 4 : 4 π

π 4

Apply the commutative law:

π 4 = 4 π

4 π

Simplify

2 πn \cdot 4 : 8 πn

2 πn \cdot 4

Multiply the numbers:

2 \cdot 4 = 8

= 8 πn

8 x - x = 4 π + 8 πn

7 x = 4 π + 8 πn

7 x = 4 π + 8 πn

7 x = 4 π + 8 πn

Divide both sides by

7

7 x = 4 π + 8 πn

Divide both sides by

7

\frac{7 x}{7} = \frac{4 π}{7} + \frac{8 πn}{7}

Simplify

x = \frac{4 π}{7} + \frac{8 πn}{7}

x = \frac{4 π}{7} + \frac{8 πn}{7}

x = \frac{8 πn}{7}, x = \frac{4 π}{7} + \frac{8 πn}{7}

Popular Examples

5+tan(x)=4

5 + tan (x) = 4

sin^2(x)-6sin(x)=0

sin^{2} (x) - 6 sin (x) = 0

cos(a)= 3/3

cos (a) = \frac{3}{3}

3csc^2(x)-2csc(x)-5=0

3 csc^{2} (x) - 2 csc (x) - 5 = 0

cos(2x-1)=0.4

cos (2 x - 1) = 0.4

Frequently Asked Questions (FAQ)

What is the general solution for 5^{sin(2x-x/4)}=1 ?
The general solution for 5^{sin(2x-x/4)}=1 is x=(8pin)/7 ,x=(4pi)/7+(8pin)/7