Finite Math Examples

$e^{2 x - 1} = 405$

Step 1

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

$\ln (e^{2 x - 1}) = \ln (405)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (e^{2 x - 1})$ by moving $2 x - 1$ outside the logarithm.

$(2 x - 1) \ln (e) = \ln (405)$

Step 2.2

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

$(2 x - 1) \cdot 1 = \ln (405)$

Step 2.3

Multiply $2 x - 1$ by $1$ .

$2 x - 1 = \ln (405)$

Step 3

Add $1$ to both sides of the equation.

$2 x = \ln (405) + 1$

Step 4

Divide each term in $2 x = \ln (405) + 1$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = \ln (405) + 1$ by $2$ .

$\frac{2 x}{2} = \frac{\ln (405)}{2} + \frac{1}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\ln (405)}{2} + \frac{1}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (405)}{2} + \frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (405)}{2} + \frac{1}{2}$

Decimal Form:

$x = 3.50194353 \dots$