Finite Math Examples

$e^{2 x + 10} = 5^{\frac{3 x}{11}}$

Step 1

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

$\ln (e^{2 x + 10}) = \ln (5^{\frac{3 x}{11}})$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (e^{2 x + 10})$ by moving $2 x + 10$ outside the logarithm.

$(2 x + 10) \ln (e) = \ln (5^{\frac{3 x}{11}})$

Step 2.2

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

$(2 x + 10) \cdot 1 = \ln (5^{\frac{3 x}{11}})$

Step 2.3

Multiply $2 x + 10$ by $1$ .

$2 x + 10 = \ln (5^{\frac{3 x}{11}})$

Step 3

Expand the right side.

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Step 3.1

Expand $\ln (5^{\frac{3 x}{11}})$ by moving $\frac{3 x}{11}$ outside the logarithm.

$2 x + 10 = \frac{3 x}{11} \ln (5)$

Step 3.2

Combine $\frac{3 x}{11}$ and $\ln (5)$ .

$2 x + 10 = \frac{3 x \ln (5)}{11}$

Step 4

Multiply both sides by $11$ .

$(2 x + 10) \cdot 11 = \frac{3 x \ln (5)}{11} \cdot 11$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $(2 x + 10) \cdot 11$ .

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Step 5.1.1.1

Apply the distributive property.

$2 x \cdot 11 + 10 \cdot 11 = \frac{3 x \ln (5)}{11} \cdot 11$

Step 5.1.1.2

Multiply.

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Step 5.1.1.2.1

Multiply $11$ by $2$ .

$22 x + 10 \cdot 11 = \frac{3 x \ln (5)}{11} \cdot 11$

Step 5.1.1.2.2

Multiply $10$ by $11$ .

$22 x + 110 = \frac{3 x \ln (5)}{11} \cdot 11$

Step 5.2

Simplify the right side.

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Step 5.2.1

Cancel the common factor of $11$ .

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Step 5.2.1.1

Simplify $3 \ln (5)$ by moving $3$ inside the logarithm.

$22 x + 110 = \frac{x \ln (5^{3})}{11} \cdot 11$

Step 5.2.1.2

Raise $5$ to the power of $3$ .

$22 x + 110 = \frac{x \ln (125)}{11} \cdot 11$

Step 5.2.1.3

Cancel the common factor.

$22 x + 110 = \frac{x \ln (125)}{11} \cdot 11$

Step 5.2.1.4

Rewrite the expression.

$22 x + 110 = x \ln (125)$

Step 6

Solve for $x$ .

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Step 6.1

Subtract $x \ln (125)$ from both sides of the equation.

$22 x + 110 - x \ln (125) = 0$

Step 6.2

Subtract $110$ from both sides of the equation.

$22 x - x \ln (125) = - 110$

Step 6.3

Factor the left side of the equation.

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Step 6.3.1

Factor $x$ out of $22 x - x \ln (125)$ .

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Step 6.3.1.1

Factor $x$ out of $22 x$ .

$x \cdot 22 - x \ln (125) = - 110$

Step 6.3.1.2

Factor $x$ out of $- x \ln (125)$ .

$x \cdot 22 + x (- 1 \ln (125)) = - 110$

Step 6.3.1.3

Factor $x$ out of $x \cdot 22 + x (- 1 \ln (125))$ .

$x (22 - 1 \ln (125)) = - 110$

Step 6.3.2

Rewrite $- 1 \ln (125)$ as $- \ln (125)$ .

$x (22 - \ln (125)) = - 110$

Step 6.4

Divide each term in $x (22 - \ln (125)) = - 110$ by $22 - \ln (125)$ and simplify.

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Step 6.4.1

Divide each term in $x (22 - \ln (125)) = - 110$ by $22 - \ln (125)$ .

$\frac{x (22 - \ln (125))}{22 - \ln (125)} = \frac{- 110}{22 - \ln (125)}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $22 - \ln (125)$ .

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Step 6.4.2.1.1

Cancel the common factor.

$\frac{x (22 - \ln (125))}{22 - \ln (125)} = \frac{- 110}{22 - \ln (125)}$

Step 6.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 110}{22 - \ln (125)}$

Step 6.4.3

Simplify the right side.

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Step 6.4.3.1

Move the negative in front of the fraction.

$x = - \frac{110}{22 - \ln (125)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{110}{22 - \ln (125)}$

Decimal Form:

$x = - 6.40589388 \dots$