Solve by Factoring 10^(-3x)*10^x=1/10

$10^{- 3 x} \cdot 10^{x} = \frac{1}{10}$

Step 1

Subtract $\frac{1}{10}$ from both sides of the equation.

$10^{- 3 x} \cdot 10^{x} - \frac{1}{10} = 0$

Step 2

Multiply $10^{- 3 x}$ by $10^{x}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$10^{- 3 x + x} - \frac{1}{10} = 0$

Add $- 3 x$ and $x$ .

$10^{- 2 x} - \frac{1}{10} = 0$

Step 3

To write $10^{- 2 x}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$10^{- 2 x} \cdot \frac{10}{10} - \frac{1}{10} = 0$

Step 4

Combine $10^{- 2 x}$ and $\frac{10}{10}$ .

$\frac{10^{- 2 x} \cdot 10}{10} - \frac{1}{10} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{10^{- 2 x} \cdot 10 - 1}{10} = 0$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{10^{- 2 x + 1} - 1}{10} = 0$

Step 7

Set the numerator equal to zero.

$10^{- 2 x + 1} - 1 = 0$

Step 8

Solve the equation for $x$ .

Add $1$ to both sides of the equation.

$10^{- 2 x + 1} = 1$

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{- 2 x + 1}) = \log (1)$

Expand the left side.

Expand $\log (10^{- 2 x + 1})$ by moving $- 2 x + 1$ outside the logarithm.

$(- 2 x + 1) \log (10) = \log (1)$

Logarithm base $10$ of $10$ is $1$ .

$(- 2 x + 1) \cdot 1 = \log (1)$

Multiply $- 2 x + 1$ by $1$ .

$- 2 x + 1 = \log (1)$

Simplify the right side.

Logarithm base $10$ of $1$ is $0$ .

$- 2 x + 1 = 0$

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Divide each term in $- 2 x = - 1$ by $- 2$ and simplify.

Divide each term in $- 2 x = - 1$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 1}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 1}{- 2}$

Divide $x$ by $1$ .

$x = \frac{- 1}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{1}{2}$

Solve by Factoring 10^(-3x)*10^x=1/10

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