Solve for m -6.5e^(m-3)=-68

$- 6.5 e^{m - 3} = - 68$

Step 1

Divide each term in $- 6.5 e^{m - 3} = - 68$ by $- 6.5$ and simplify.

Divide each term in $- 6.5 e^{m - 3} = - 68$ by $- 6.5$ .

$\frac{- 6.5 e^{m - 3}}{- 6.5} = \frac{- 68}{- 6.5}$

Simplify the left side.

Cancel the common factor of $- 6.5$ .

Cancel the common factor.

$\frac{- 6.5 e^{m - 3}}{- 6.5} = \frac{- 68}{- 6.5}$

Divide $e^{m - 3}$ by $1$ .

$e^{m - 3} = \frac{- 68}{- 6.5}$

Simplify the right side.

Divide $- 68$ by $- 6.5$ .

$e^{m - 3} = 10.\overline{461538}$

Step 2

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

$\ln (e^{m - 3}) = \ln (10.\overline{461538})$

Step 3

Expand the left side.

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

$(m - 3) \ln (e) = \ln (10.\overline{461538})$

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

$(m - 3) \cdot 1 = \ln (10.\overline{461538})$

Multiply $m - 3$ by $1$ .

$m - 3 = \ln (10.\overline{461538})$

Step 4

Add $3$ to both sides of the equation.

$m = \ln (10.\overline{461538}) + 3$

Step 5

The result can be shown in multiple forms.

Exact Form:

$m = \ln (10.\overline{461538}) + 3$

Decimal Form:

$m = 5.34770552 \dots$