Finite Math Examples

$(1.5) \cdot 2^{3 x} = 2 e$

Step 1

Divide each term in $1.5 \cdot 2^{3 x} = 2 e$ by $1.5$ and simplify.

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Step 1.1

Divide each term in $1.5 \cdot 2^{3 x} = 2 e$ by $1.5$ .

$\frac{1.5 \cdot 2^{3 x}}{1.5} = \frac{2 e}{1.5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $1.5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{1.5 \cdot 2^{3 x}}{1.5} = \frac{2 e}{1.5}$

Step 1.2.1.2

Divide $2^{3 x}$ by $1$ .

$2^{3 x} = \frac{2 e}{1.5}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Replace $e$ with an approximation.

$2^{3 x} = \frac{2 \cdot 2.71828182}{1.5}$

Step 1.3.2

Multiply $2$ by $2.71828182$ .

$2^{3 x} = \frac{5.43656365}{1.5}$

Step 1.3.3

Divide $5.43656365$ by $1.5$ .

$2^{3 x} = 3.62437577$

Step 2

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

$\ln (2^{3 x}) = \ln (3.62437577)$

Step 3

Expand $\ln (2^{3 x})$ by moving $3 x$ outside the logarithm.

$3 x \ln (2) = \ln (3.62437577)$

Step 4

Divide each term in $3 x \ln (2) = \ln (3.62437577)$ by $3 \ln (2)$ and simplify.

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Step 4.1

Divide each term in $3 x \ln (2) = \ln (3.62437577)$ by $3 \ln (2)$ .

$\frac{3 x \ln (2)}{3 \ln (2)} = \frac{\ln (3.62437577)}{3 \ln (2)}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 x \ln (2)}{3 \ln (2)} = \frac{\ln (3.62437577)}{3 \ln (2)}$

Step 4.2.1.2

Rewrite the expression.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (3.62437577)}{3 \ln (2)}$

Step 4.2.2

Cancel the common factor of $\ln (2)$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (3.62437577)}{3 \ln (2)}$

Step 4.2.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (3.62437577)}{3 \ln (2)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (3.62437577)}{3 \ln (2)}$

Decimal Form:

$x = 0.61924418 \dots$