Finite Math Examples

$\ln (x) = \frac{1}{2} \cdot \ln (2 x + \frac{5}{2}) + \frac{1}{2} \cdot \ln (2)$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\frac{1}{2} \cdot \ln (2 x + \frac{5}{2})$ from both sides of the equation.

$\ln (x) - \frac{1}{2} \cdot \ln (2 x + \frac{5}{2}) = \frac{1}{2} \cdot \ln (2)$

Step 1.2

Subtract $\frac{1}{2} \cdot \ln (2)$ from both sides of the equation.

$\ln (x) - \frac{1}{2} \cdot \ln (2 x + \frac{5}{2}) - \frac{1}{2} \cdot \ln (2) = 0$

Step 2

Simplify $\ln (x) - \frac{1}{2} \cdot \ln (2 x + \frac{5}{2}) - \frac{1}{2} \cdot \ln (2)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $- \frac{1}{2} \ln (2 x + \frac{5}{2})$ .

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Step 2.1.1.1

Reorder $\ln (2 x + \frac{5}{2})$ and $\frac{1}{2}$ .

$\ln (x) - (\frac{1}{2} \ln (2 x + \frac{5}{2})) - \frac{1}{2} \cdot \ln (2) = 0$

Step 2.1.1.2

Simplify $\frac{1}{2} \ln (2 x + \frac{5}{2})$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln (x) - \ln ({(2 x + \frac{5}{2})}^{\frac{1}{2}}) - \frac{1}{2} \cdot \ln (2) = 0$

Step 2.1.2

Multiply $- \frac{1}{2} \ln (2)$ .

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Step 2.1.2.1

Reorder $\ln (2)$ and $\frac{1}{2}$ .

$\ln (x) - \ln ({(2 x + \frac{5}{2})}^{\frac{1}{2}}) - (\frac{1}{2} \ln (2)) = 0$

Step 2.1.2.2

Simplify $\frac{1}{2} \ln (2)$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln (x) - \ln ({(2 x + \frac{5}{2})}^{\frac{1}{2}}) - \ln (2^{\frac{1}{2}}) = 0$

Step 2.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{x}{{(2 x + \frac{5}{2})}^{\frac{1}{2}}}) - \ln (2^{\frac{1}{2}}) = 0$

Step 2.3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{\frac{x}{{(2 x + \frac{5}{2})}^{\frac{1}{2}}}}{2^{\frac{1}{2}}}) = 0$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{x}{{(2 x + \frac{5}{2})}^{\frac{1}{2}}} \cdot \frac{1}{2^{\frac{1}{2}}}) = 0$

Step 2.5

Combine.

$\ln (\frac{x \cdot 1}{{(2 x + \frac{5}{2})}^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}}) = 0$

Step 2.6

Multiply $x$ by $1$ .

$\ln (\frac{x}{{(2 x + \frac{5}{2})}^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}}) = 0$

Step 2.7

Simplify the denominator.

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Step 2.7.1

To write $2 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\ln (\frac{x}{{(2 x \cdot \frac{2}{2} + \frac{5}{2})}^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}}) = 0$

Step 2.7.2

Combine $2 x$ and $\frac{2}{2}$ .

$\ln (\frac{x}{{(\frac{2 x \cdot 2}{2} + \frac{5}{2})}^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}}) = 0$

Step 2.7.3

Combine the numerators over the common denominator.

$\ln (\frac{x}{{(\frac{2 x \cdot 2 + 5}{2})}^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}}) = 0$

Step 2.7.4

Multiply $2$ by $2$ .

$\ln (\frac{x}{{(\frac{4 x + 5}{2})}^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}}) = 0$

Step 2.7.5

Apply the product rule to $\frac{4 x + 5}{2}$ .

$\ln (\frac{x}{\frac{{(4 x + 5)}^{\frac{1}{2}}}{2^{\frac{1}{2}}} \cdot 2^{\frac{1}{2}}}) = 0$

Step 2.8

Combine $\frac{{(4 x + 5)}^{\frac{1}{2}}}{2^{\frac{1}{2}}}$ and $2^{\frac{1}{2}}$ .

$\ln (\frac{x}{\frac{{(4 x + 5)}^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}}{2^{\frac{1}{2}}}}) = 0$

Step 2.9

Reduce the expression by cancelling the common factors.

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Step 2.9.1

Reduce the expression $\frac{{(4 x + 5)}^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}}{2^{\frac{1}{2}}}$ by cancelling the common factors.

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Step 2.9.1.1

Cancel the common factor.

$\ln (\frac{x}{\frac{{(4 x + 5)}^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}}{2^{\frac{1}{2}}}}) = 0$

Step 2.9.1.2

Rewrite the expression.

$\ln (\frac{x}{\frac{{(4 x + 5)}^{\frac{1}{2}}}{1}}) = 0$

Step 2.9.2

Divide ${(4 x + 5)}^{\frac{1}{2}}$ by $1$ .

$\ln (\frac{x}{{(4 x + 5)}^{\frac{1}{2}}}) = 0$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 5$

Step 4