Solve for x (m^(x-7))/(m^-3)=(m^2)(m^-2)

$\frac{m^{x - 7}}{m^{- 3}} = (m^{2}) (m^{- 2})$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (\frac{m^{x - 7}}{m^{- 3}}) = \ln (m^{2} (m^{- 2}))$

Step 2

Expand the left side.

Rewrite $\ln (\frac{m^{x - 7}}{m^{- 3}})$ as $\ln (m^{x - 7}) - \ln (m^{- 3})$ .

$\ln (m^{x - 7}) - \ln (m^{- 3}) = \ln (m^{2} (m^{- 2}))$

Expand $\ln (m^{x - 7})$ by moving $x - 7$ outside the logarithm.

$(x - 7) \ln (m) - \ln (m^{- 3}) = \ln (m^{2} (m^{- 2}))$

Expand $\ln (m^{- 3})$ by moving $- 3$ outside the logarithm.

$(x - 7) \ln (m) - (- 3 \ln (m)) = \ln (m^{2} (m^{- 2}))$

Multiply $- 3$ by $- 1$ .

$(x - 7) \ln (m) + 3 \ln (m) = \ln (m^{2} (m^{- 2}))$

Step 3

Simplify the left side.

Simplify $(x - 7) \ln (m) + 3 \ln (m)$ .

Apply the distributive property.

$x \ln (m) - 7 \ln (m) + 3 \ln (m) = \ln (m^{2} (m^{- 2}))$

Add $- 7 \ln (m)$ and $3 \ln (m)$ .

$x \ln (m) - 4 \ln (m) = \ln (m^{2} (m^{- 2}))$

Step 4

Simplify the right side.

Simplify $\ln (m^{2} (m^{- 2}))$ .

Multiply $m^{2}$ by $m^{- 2}$ by adding the exponents.

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

$x \ln (m) - 4 \ln (m) = \ln (m^{2 - 2})$

Subtract $2$ from $2$ .

$x \ln (m) - 4 \ln (m) = \ln (m^{0})$

Simplify $m^{0}$ .

$x \ln (m) - 4 \ln (m) = \ln (1)$

The natural logarithm of $1$ is $0$ .

$x \ln (m) - 4 \ln (m) = 0$

Step 5

Add $4 \ln (m)$ to both sides of the equation.

$x \ln (m) = 4 \ln (m)$

Step 6

Divide each term in $x \ln (m) = 4 \ln (m)$ by $\ln (m)$ and simplify.

Divide each term in $x \ln (m) = 4 \ln (m)$ by $\ln (m)$ .

$\frac{x \ln (m)}{\ln (m)} = \frac{4 \ln (m)}{\ln (m)}$

Simplify the left side.

Cancel the common factor of $\ln (m)$ .

Cancel the common factor.

$\frac{x \ln (m)}{\ln (m)} = \frac{4 \ln (m)}{\ln (m)}$

Divide $x$ by $1$ .

$x = \frac{4 \ln (m)}{\ln (m)}$

Simplify the right side.

Cancel the common factor of $\ln (m)$ .

Cancel the common factor.

$x = \frac{4 \ln (m)}{\ln (m)}$

Divide $4$ by $1$ .

$x = 4$

Solve for x (m^(x-7))/(m^-3)=(m^2)(m^-2)

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