Solve for x 10^(x-3)=100^(4x-5)

$10^{x - 3} = 100^{4 x - 5}$

Step 1

Create equivalent expressions in the equation that all have equal bases.

$10^{x - 3} = 10^{2 (4 x - 5)}$

Step 2

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

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

Step 3

Solve for $x$ .

Simplify $2 (4 x - 5)$ .

Rewrite.

$x - 3 = 0 + 0 + 2 (4 x - 5)$

Simplify by adding zeros.

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

Apply the distributive property.

$x - 3 = 2 (4 x) + 2 \cdot - 5$

Multiply.

Multiply $4$ by $2$ .

$x - 3 = 8 x + 2 \cdot - 5$

Multiply $2$ by $- 5$ .

$x - 3 = 8 x - 10$

Move all terms containing $x$ to the left side of the equation.

Subtract $8 x$ from both sides of the equation.

$x - 3 - 8 x = - 10$

Subtract $8 x$ from $x$ .

$- 7 x - 3 = - 10$

Move all terms not containing $x$ to the right side of the equation.

Add $3$ to both sides of the equation.

$- 7 x = - 10 + 3$

Add $- 10$ and $3$ .

$- 7 x = - 7$

Divide each term in $- 7 x = - 7$ by $- 7$ and simplify.

Divide each term in $- 7 x = - 7$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{- 7}{- 7}$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{- 7}{- 7}$

Divide $x$ by $1$ .

$x = \frac{- 7}{- 7}$

Simplify the right side.

Divide $- 7$ by $- 7$ .

$x = 1$