Finite Math Examples

$\frac{10 e^{3 z - 7}}{10} = \frac{5}{10}$

Step 1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$10 e^{3 z - 7} = 5$

Step 2

Divide each term in $10 e^{3 z - 7} = 5$ by $10$ and simplify.

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Step 2.1

Divide each term in $10 e^{3 z - 7} = 5$ by $10$ .

$\frac{10 e^{3 z - 7}}{10} = \frac{5}{10}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $10$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{10 e^{3 z - 7}}{10} = \frac{5}{10}$

Step 2.2.1.2

Divide $e^{3 z - 7}$ by $1$ .

$e^{3 z - 7} = \frac{5}{10}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $5$ and $10$ .

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Step 2.3.1.1

Factor $5$ out of $5$ .

$e^{3 z - 7} = \frac{5 (1)}{10}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $5$ out of $10$ .

$e^{3 z - 7} = \frac{5 \cdot 1}{5 \cdot 2}$

Step 2.3.1.2.2

Cancel the common factor.

$e^{3 z - 7} = \frac{5 \cdot 1}{5 \cdot 2}$

Step 2.3.1.2.3

Rewrite the expression.

$e^{3 z - 7} = \frac{1}{2}$

Step 3

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

$\ln (e^{3 z - 7}) = \ln (\frac{1}{2})$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{3 z - 7})$ by moving $3 z - 7$ outside the logarithm.

$(3 z - 7) \ln (e) = \ln (\frac{1}{2})$

Step 4.2

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

$(3 z - 7) \cdot 1 = \ln (\frac{1}{2})$

Step 4.3

Multiply $3 z - 7$ by $1$ .

$3 z - 7 = \ln (\frac{1}{2})$

Step 5

Add $7$ to both sides of the equation.

$3 z = \ln (\frac{1}{2}) + 7$

Step 6

Divide each term in $3 z = \ln (\frac{1}{2}) + 7$ by $3$ and simplify.

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Step 6.1

Divide each term in $3 z = \ln (\frac{1}{2}) + 7$ by $3$ .

$\frac{3 z}{3} = \frac{\ln (\frac{1}{2})}{3} + \frac{7}{3}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $3$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{3 z}{3} = \frac{\ln (\frac{1}{2})}{3} + \frac{7}{3}$

Step 6.2.1.2

Divide $z$ by $1$ .

$z = \frac{\ln (\frac{1}{2})}{3} + \frac{7}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$z = \frac{\ln (\frac{1}{2})}{3} + \frac{7}{3}$

Decimal Form:

$z = 2.10228427 \dots$