Solve y=3 log_(5) (4x^(4))-6

Question

Function

Evaluate the derivative

Find the domain

Find the x -intercept/zero

y^{'} = \frac{12}{ln ( 5 ) \times x}

Evaluate

y = 3 lo g_{5} (4 x^{4}) - 6

Take the derivative of both sides

y^{'} = \frac{d}{d x} (3 lo g_{5} (4 x^{4}) - 6)

Use differentiation rule \frac{d}{d x} (f (x) \pm g (x)) = \frac{d}{d x} (f (x)) \pm \frac{d}{d x} (g (x))

y^{'} = \frac{d}{d x} (3 lo g_{5} (4 x^{4})) - \frac{d}{d x} (6)

Calculate

More Steps

Calculate

\frac{d}{d x} (3 lo g_{5} (4 x^{4}))

Simplify

3 \times \frac{d}{d x} (lo g_{5} (4 x^{4}))

Calculate

3 \times \frac{4}{ln ( 5 ) \times x}

Multiply the terms

\frac{3 \times 4}{ln ( 5 ) \times x}

Multiply the terms

\frac{12}{ln ( 5 ) \times x}

y^{'} = \frac{12}{ln ( 5 ) \times x} - \frac{d}{d x} (6)

Use \frac{d}{d x} (c) = 0 to find derivative

y^{'} = \frac{12}{ln ( 5 ) \times x} - 0

Solution

y^{'} = \frac{12}{ln ( 5 ) \times x}

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Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Not symmetry with respect to the origin

Evaluate

y = 3 lo g_{5} (4 x^{4}) - 6

To test if the graph of y = 3 lo g_{5} (4 x^{4}) - 6 is symmetry with respect to the origin,substitute -x for x and -y for y

- y = 3 lo g_{5} (4 (- x)^{4}) - 6

Simplify

- y = 3 lo g_{5} (4 x^{4}) - 6

Change the signs both sides

y = - 3 lo g_{5} (4 x^{4}) + 6

Solution

Not symmetry with respect to the origin

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{4 2 \times 5 ^{\frac{6 + y}{12}}}{2} x = - \frac{4 2 \times 5 ^{\frac{6 + y}{12}}}{2}

Evaluate

y = 3 lo g_{5} (4 x^{4}) - 6

Swap the sides of the equation

3 lo g_{5} (4 x^{4}) - 6 = y

Move the expression to the left side

3 lo g_{5} (4 x^{4}) - 6 - y = 0

Solve the equation using substitution t = lo g_{5} (4 x^{4})

3 t - 6 - y = 0

Move the expression to the right-hand side and change its sign

3 t = 0 + 6 + y

Removing 0 doesn't change the value,so remove it from the expression

3 t = 6 + y

Divide both sides

\frac{3 t}{3} = \frac{6 + y}{3}

Divide the numbers

t = \frac{6 + y}{3}

Substitute back

lo g_{5} (4 x^{4}) = \frac{6 + y}{3}

Convert the logarithm into exponential form using the fact that lo g_{a} x = b is equal to x = a^{b}

4 x^{4} = 5^{\frac{6 + y}{3}}

Divide both sides

\frac{4 x ^{4}}{4} = \frac{5 ^{\frac{6 + y}{3}}}{4}

Divide the numbers

x^{4} = \frac{5 ^{\frac{6 + y}{3}}}{4}

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm 4 \frac{5 ^{\frac{6 + y}{3}}}{4}

Simplify the expression

More Steps

Evaluate

4 \frac{5 ^{\frac{6 + y}{3}}}{4}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{4 5 ^{\frac{6 + y}{3}}}{4 4}

Simplify the radical expression

More Steps

Evaluate

44

Write the number in exponential form with the base of 2

4 2^{2}

Reduce the index of the radical and exponent with 2

2

\frac{4 5 ^{\frac{6 + y}{3}}}{2}

Multiply by the Conjugate

\frac{4 5 ^{\frac{6 + y}{3}} \times 2}{2 \times 2}

Calculate

\frac{4 5 ^{\frac{6 + y}{3}} \times 2}{2}

Calculate

More Steps

Evaluate

4 5^{\frac{6 + y}{3}} \times 2

Use n a = mn a^{m} to expand the expression

4 5^{\frac{6 + y}{3}} \times 44

The product of roots with the same index is equal to the root of the product

4 5^{\frac{6 + y}{3}} \times 4

Calculate the product

4 4 \times 5^{\frac{6 + y}{3}}

\frac{4 4 \times 5 ^{\frac{6 + y}{3}}}{2}

x = \pm \frac{4 4 \times 5 ^{\frac{6 + y}{3}}}{2}

Separate the equation into 2 possible cases

x = \frac{4 4 \times 5 ^{\frac{6 + y}{3}}}{2} x = - \frac{4 4 \times 5 ^{\frac{6 + y}{3}}}{2}

Simplify

x = \frac{4 2 \times 5 ^{\frac{6 + y}{12}}}{2} x = - \frac{4 4 \times 5 ^{\frac{6 + y}{3}}}{2}

Solution

x = \frac{4 2 \times 5 ^{\frac{6 + y}{12}}}{2} x = - \frac{4 2 \times 5 ^{\frac{6 + y}{12}}}{2}

Show Solution

Function

Testing for symmetry

Solve the equation

Graph