Solve ( 625/4096 )^(5x-1)=( 512/125 )^(x-2)

Question

Solve the equation

x = \frac{10}{23}

Alternative Form

x \approx 0.434783

Evaluate

(\frac{625}{4096})^{5 x - 1} = (\frac{512}{125})^{x - 2}

Take the logarithm of both sides

lo g_{\frac{625}{4096}} ((\frac{625}{4096})^{5 x - 1}) = lo g_{\frac{625}{4096}} ((\frac{512}{125})^{x - 2})

Evaluate the logarithm

\frac{4 ( 5 x - 1 )}{4} = - \frac{3 ( x - 2 )}{4}

Rewrite the expression

\frac{4 ( 5 x - 1 )}{4} = \frac{- 3 ( x - 2 )}{4}

Cross multiply

4 (5 x - 1) \times 4 = 4 (- 3 (x - 2))

Simplify the equation

16 (5 x - 1) = 4 (- 3 (x - 2))

Simplify the equation

16 (5 x - 1) = - 12 (x - 2)

Rewrite the expression

4 \times 4 (5 x - 1) = 4 (- 3 (x - 2))

Evaluate

4 (5 x - 1) = - 3 (x - 2)

Calculate

More Steps

Evaluate

4 (5 x - 1)

Apply the distributive property

4 \times 5 x - 4 \times 1

Multiply the numbers

20 x - 4 \times 1

Any expression multiplied by 1 remains the same

20 x - 4

20 x - 4 = - 3 (x - 2)

Calculate

More Steps

Evaluate

- 3 (x - 2)

Apply the distributive property

- 3 x - (- 3 \times 2)

Multiply the numbers

- 3 x - (- 6)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

- 3 x + 6

20 x - 4 = - 3 x + 6

Move the expression to the left side

20 x - 4 - (- 3 x + 6) = 0

Calculate

More Steps

Add the terms

20 x - 4 - (- 3 x + 6)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

20 x - 4 + 3 x - 6

Add the terms

More Steps

Evaluate

20 x + 3 x

Collect like terms by calculating the sum or difference of their coefficients

(20 + 3) x

Add the numbers

23 x

23 x - 4 - 6

Subtract the numbers

23 x - 10

23 x - 10 = 0

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

23 x = 0 + 10

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

23 x = 10

Divide both sides

\frac{23 x}{23} = \frac{10}{23}

Solution

x = \frac{10}{23}

Alternative Form

x \approx 0.434783

Show Solution

Solve the equation

Graph