Solve for y 2/7*e^(6y-5)=14

$\frac{2}{7} \cdot e^{6 y - 5} = 14$

Step 1

Multiply both sides of the equation by $\frac{7}{2}$ .

$\frac{7}{2} (\frac{2}{7} \cdot e^{6 y - 5}) = \frac{7}{2} \cdot 14$

Step 2

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{7}{2} (\frac{2}{7} \cdot e^{6 y - 5})$ .

Combine $\frac{2}{7}$ and $e^{6 y - 5}$ .

$\frac{7}{2} \cdot \frac{2 e^{6 y - 5}}{7} = \frac{7}{2} \cdot 14$

Combine.

$\frac{7 (2 e^{6 y - 5})}{2 \cdot 7} = \frac{7}{2} \cdot 14$

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 (2 e^{6 y - 5})}{2 \cdot 7} = \frac{7}{2} \cdot 14$

Rewrite the expression.

$\frac{2 e^{6 y - 5}}{2} = \frac{7}{2} \cdot 14$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 e^{6 y - 5}}{2} = \frac{7}{2} \cdot 14$

Divide $e^{6 y - 5}$ by $1$ .

$e^{6 y - 5} = \frac{7}{2} \cdot 14$

Simplify the right side.

Simplify $\frac{7}{2} \cdot 14$ .

Cancel the common factor of $2$ .

Factor $2$ out of $14$ .

$e^{6 y - 5} = \frac{7}{2} \cdot (2 (7))$

Cancel the common factor.

$e^{6 y - 5} = \frac{7}{2} \cdot (2 \cdot 7)$

Rewrite the expression.

$e^{6 y - 5} = 7 \cdot 7$

Multiply $7$ by $7$ .

$e^{6 y - 5} = 49$

Step 3

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

$\ln (e^{6 y - 5}) = \ln (49)$

Step 4

Expand the left side.

Expand $\ln (e^{6 y - 5})$ by moving $6 y - 5$ outside the logarithm.

$(6 y - 5) \ln (e) = \ln (49)$

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

$(6 y - 5) \cdot 1 = \ln (49)$

Multiply $6 y - 5$ by $1$ .

$6 y - 5 = \ln (49)$

Step 5

Add $5$ to both sides of the equation.

$6 y = \ln (49) + 5$

Step 6

Divide each term in $6 y = \ln (49) + 5$ by $6$ and simplify.

Divide each term in $6 y = \ln (49) + 5$ by $6$ .

$\frac{6 y}{6} = \frac{\ln (49)}{6} + \frac{5}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{\ln (49)}{6} + \frac{5}{6}$

Divide $y$ by $1$ .

$y = \frac{\ln (49)}{6} + \frac{5}{6}$

Simplify the right side.

Simplify each term.

Rewrite $\ln (49)$ as $\ln (7^{2})$ .

$y = \frac{\ln (7^{2})}{6} + \frac{5}{6}$

Expand $\ln (7^{2})$ by moving $2$ outside the logarithm.

$y = \frac{2 \ln (7)}{6} + \frac{5}{6}$

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2 \ln (7)$ .

$y = \frac{2 (\ln (7))}{6} + \frac{5}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$y = \frac{2 \ln (7)}{2 \cdot 3} + \frac{5}{6}$

Cancel the common factor.

$y = \frac{2 \ln (7)}{2 \cdot 3} + \frac{5}{6}$

Rewrite the expression.

$y = \frac{\ln (7)}{3} + \frac{5}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = \frac{\ln (7)}{3} + \frac{5}{6}$

Decimal Form:

$y = 1.48197004 \dots$

Solve for y 2/7*e^(6y-5)=14

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