Algebra Examples

$5 e^{4 m - 3} - 7 = 13$

Step 1

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

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Step 1.1

Add $7$ to both sides of the equation.

$5 e^{4 m - 3} = 13 + 7$

Step 1.2

Add $13$ and $7$ .

$5 e^{4 m - 3} = 20$

Step 2

Divide each term in $5 e^{4 m - 3} = 20$ by $5$ and simplify.

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Step 2.1

Divide each term in $5 e^{4 m - 3} = 20$ by $5$ .

$\frac{5 e^{4 m - 3}}{5} = \frac{20}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $5$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{5 e^{4 m - 3}}{5} = \frac{20}{5}$

Step 2.2.1.2

Divide $e^{4 m - 3}$ by $1$ .

$e^{4 m - 3} = \frac{20}{5}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $20$ by $5$ .

$e^{4 m - 3} = 4$

Step 3

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

$\ln (e^{4 m - 3}) = \ln (4)$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{4 m - 3})$ by moving $4 m - 3$ outside the logarithm.

$(4 m - 3) \ln (e) = \ln (4)$

Step 4.2

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

$(4 m - 3) \cdot 1 = \ln (4)$

Step 4.3

Multiply $4 m - 3$ by $1$ .

$4 m - 3 = \ln (4)$

Step 5

Add $3$ to both sides of the equation.

$4 m = \ln (4) + 3$

Step 6

Divide each term in $4 m = \ln (4) + 3$ by $4$ and simplify.

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Step 6.1

Divide each term in $4 m = \ln (4) + 3$ by $4$ .

$\frac{4 m}{4} = \frac{\ln (4)}{4} + \frac{3}{4}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $4$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{4 m}{4} = \frac{\ln (4)}{4} + \frac{3}{4}$

Step 6.2.1.2

Divide $m$ by $1$ .

$m = \frac{\ln (4)}{4} + \frac{3}{4}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$m = \frac{\ln (2^{2})}{4} + \frac{3}{4}$

Step 6.3.1.2

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

$m = \frac{2 \ln (2)}{4} + \frac{3}{4}$

Step 6.3.1.3

Cancel the common factor of $2$ and $4$ .

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Step 6.3.1.3.1

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

$m = \frac{2 (\ln (2))}{4} + \frac{3}{4}$

Step 6.3.1.3.2

Cancel the common factors.

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Step 6.3.1.3.2.1

Factor $2$ out of $4$ .

$m = \frac{2 \ln (2)}{2 \cdot 2} + \frac{3}{4}$

Step 6.3.1.3.2.2

Cancel the common factor.

$m = \frac{2 \ln (2)}{2 \cdot 2} + \frac{3}{4}$

Step 6.3.1.3.2.3

Rewrite the expression.

$m = \frac{\ln (2)}{2} + \frac{3}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$m = \frac{\ln (2)}{2} + \frac{3}{4}$

Decimal Form:

$m = 1.09657359 \dots$