Algebra Examples

$e^{- 10 a - 4} = 45$

Step 1

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

$\ln (e^{- 10 a - 4}) = \ln (45)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (e^{- 10 a - 4})$ by moving $- 10 a - 4$ outside the logarithm.

$(- 10 a - 4) \ln (e) = \ln (45)$

Step 2.2

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

$(- 10 a - 4) \cdot 1 = \ln (45)$

Step 2.3

Multiply $- 10 a - 4$ by $1$ .

$- 10 a - 4 = \ln (45)$

Step 3

Add $4$ to both sides of the equation.

$- 10 a = \ln (45) + 4$

Step 4

Divide each term in $- 10 a = \ln (45) + 4$ by $- 10$ and simplify.

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Step 4.1

Divide each term in $- 10 a = \ln (45) + 4$ by $- 10$ .

$\frac{- 10 a}{- 10} = \frac{\ln (45)}{- 10} + \frac{4}{- 10}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 10$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 10 a}{- 10} = \frac{\ln (45)}{- 10} + \frac{4}{- 10}$

Step 4.2.1.2

Divide $a$ by $1$ .

$a = \frac{\ln (45)}{- 10} + \frac{4}{- 10}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Move the negative in front of the fraction.

$a = - \frac{\ln (45)}{10} + \frac{4}{- 10}$

Step 4.3.1.2

Cancel the common factor of $4$ and $- 10$ .

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Step 4.3.1.2.1

Factor $2$ out of $4$ .

$a = - \frac{\ln (45)}{10} + \frac{2 (2)}{- 10}$

Step 4.3.1.2.2

Cancel the common factors.

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Step 4.3.1.2.2.1

Factor $2$ out of $- 10$ .

$a = - \frac{\ln (45)}{10} + \frac{2 \cdot 2}{2 \cdot - 5}$

Step 4.3.1.2.2.2

Cancel the common factor.

$a = - \frac{\ln (45)}{10} + \frac{2 \cdot 2}{2 \cdot - 5}$

Step 4.3.1.2.2.3

Rewrite the expression.

$a = - \frac{\ln (45)}{10} + \frac{2}{- 5}$

Step 4.3.1.3

Move the negative in front of the fraction.

$a = - \frac{\ln (45)}{10} - \frac{2}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{\ln (45)}{10} - \frac{2}{5}$

Decimal Form:

$a = - 0.78066624 \dots$