Trigonometry Examples

$12^{2 x + 5} = 55 (7^{3 x})$

Step 1

Take the log of both sides of the equation.

$\ln (12^{2 x + 5}) = \ln (55 \cdot 7^{3 x})$

Step 2

Expand $\ln (12^{2 x + 5})$ by moving $2 x + 5$ outside the logarithm.

$(2 x + 5) \ln (12) = \ln (55 \cdot 7^{3 x})$

Step 3

Rewrite $\ln (55 \cdot 7^{3 x})$ as $\ln (55) + \ln (7^{3 x})$ .

$(2 x + 5) \ln (12) = \ln (55) + \ln (7^{3 x})$

Step 4

Expand $\ln (7^{3 x})$ by moving $3 x$ outside the logarithm.

$(2 x + 5) \ln (12) = \ln (55) + 3 x \ln (7)$

Step 5

Solve the equation for $x$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $(2 x + 5) \ln (12)$ .

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Step 5.1.1.1

Apply the distributive property.

$2 x \ln (12) + 5 \ln (12) = \ln (55) + 3 x \ln (7)$

Step 5.1.1.2

Simplify $2 \ln (12)$ by moving $2$ inside the logarithm.

$x \ln (12^{2}) + 5 \ln (12) = \ln (55) + 3 x \ln (7)$

Step 5.1.1.3

Simplify $5 \ln (12)$ by moving $5$ inside the logarithm.

$x \ln (12^{2}) + \ln (12^{5}) = \ln (55) + 3 x \ln (7)$

Step 5.1.1.4

Simplify each term.

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Step 5.1.1.4.1

Raise $12$ to the power of $2$ .

$x \ln (144) + \ln (12^{5}) = \ln (55) + 3 x \ln (7)$

Step 5.1.1.4.2

Raise $12$ to the power of $5$ .

$x \ln (144) + \ln (248832) = \ln (55) + 3 x \ln (7)$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Simplify $3 \ln (7)$ by moving $3$ inside the logarithm.

$x \ln (144) + \ln (248832) = \ln (55) + x \ln (7^{3})$

Step 5.2.1.2

Raise $7$ to the power of $3$ .

$x \ln (144) + \ln (248832) = \ln (55) + x \ln (343)$

Step 5.3

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (144) + \ln (248832) - \ln (55) - x \ln (343) = 0$

Step 5.4

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

$x \ln (144) + \ln (\frac{248832}{55}) - x \ln (343) = 0$

Step 5.5

Subtract $\ln (\frac{248832}{55})$ from both sides of the equation.

$x \ln (144) - x \ln (343) = - \ln (\frac{248832}{55})$

Step 5.6

Factor $x$ out of $x \ln (144) - x \ln (343)$ .

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Step 5.6.1

Factor $x$ out of $x \ln (144)$ .

$x (\ln (144)) - x \ln (343) = - \ln (\frac{248832}{55})$

Step 5.6.2

Factor $x$ out of $- x \ln (343)$ .

$x (\ln (144)) + x (- 1 \ln (343)) = - \ln (\frac{248832}{55})$

Step 5.6.3

Factor $x$ out of $x (\ln (144)) + x (- 1 \ln (343))$ .

$x (\ln (144) - 1 \ln (343)) = - \ln (\frac{248832}{55})$

Step 5.7

Rewrite $- 1 \ln (343)$ as $- \ln (343)$ .

$x (\ln (144) - \ln (343)) = - \ln (\frac{248832}{55})$

Step 5.8

Divide each term in $x (\ln (144) - \ln (343)) = - \ln (\frac{248832}{55})$ by $\ln (144) - \ln (343)$ and simplify.

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Step 5.8.1

Divide each term in $x (\ln (144) - \ln (343)) = - \ln (\frac{248832}{55})$ by $\ln (144) - \ln (343)$ .

$\frac{x (\ln (144) - \ln (343))}{\ln (144) - \ln (343)} = \frac{- \ln (\frac{248832}{55})}{\ln (144) - \ln (343)}$

Step 5.8.2

Simplify the left side.

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Step 5.8.2.1

Cancel the common factor of $\ln (144) - \ln (343)$ .

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Step 5.8.2.1.1

Cancel the common factor.

$\frac{x (\ln (144) - \ln (343))}{\ln (144) - \ln (343)} = \frac{- \ln (\frac{248832}{55})}{\ln (144) - \ln (343)}$

Step 5.8.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{248832}{55})}{\ln (144) - \ln (343)}$

Step 5.8.3

Simplify the right side.

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Step 5.8.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{248832}{55})}{\ln (144) - \ln (343)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (\frac{248832}{55})}{\ln (144) - \ln (343)}$

Decimal Form:

$x = 9.69816080 \dots$