Algebra Examples

$\frac{1}{2} \ln (x + 37) = \ln (x^{2} - 25) - \ln (x + 5)$

Step 1

Simplify $\frac{1}{2} \ln (x + 37)$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln ({(x + 37)}^{\frac{1}{2}}) = \ln (x^{2} - 25) - \ln (x + 5)$

Step 2

Simplify $\ln (x^{2} - 25) - \ln (x + 5)$ .

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Step 2.1

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

$\ln ({(x + 37)}^{\frac{1}{2}}) = \ln (\frac{x^{2} - 25}{x + 5})$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Rewrite $25$ as $5^{2}$ .

$\ln ({(x + 37)}^{\frac{1}{2}}) = \ln (\frac{x^{2} - 5^{2}}{x + 5})$

Step 2.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$\ln ({(x + 37)}^{\frac{1}{2}}) = \ln (\frac{(x + 5) (x - 5)}{x + 5})$

Step 2.3

Cancel the common factor of $x + 5$ .

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Step 2.3.1

Cancel the common factor.

$\ln ({(x + 37)}^{\frac{1}{2}}) = \ln (\frac{(x + 5) (x - 5)}{x + 5})$

Step 2.3.2

Divide $x - 5$ by $1$ .

$\ln ({(x + 37)}^{\frac{1}{2}}) = \ln (x - 5)$

Step 3

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

$x \approx 12$

Step 4