Trigonometry Examples

$\log (x) \cdot 9 + \log (x) \cdot 27 = 5$

Step 1

Simplify $\log (x) \cdot 9 + \log (x) \cdot 27$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Multiply $\log (x) \cdot 9$ .

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Step 1.1.1.1

Reorder $\log (x)$ and $9$ .

$9 \cdot \log (x) + \log (x) \cdot 27 = 5$

Step 1.1.1.2

Simplify $9 \cdot \log (x)$ by moving $9$ inside the logarithm.

$\log (x^{9}) + \log (x) \cdot 27 = 5$

Step 1.1.2

Multiply $\log (x) \cdot 27$ .

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Step 1.1.2.1

Reorder $\log (x)$ and $27$ .

$\log (x^{9}) + 27 \cdot \log (x) = 5$

Step 1.1.2.2

Simplify $27 \cdot \log (x)$ by moving $27$ inside the logarithm.

$\log (x^{9}) + \log (x^{27}) = 5$

Step 1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (x^{9} x^{27}) = 5$

Step 1.3

Multiply $x^{9}$ by $x^{27}$ by adding the exponents.

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Step 1.3.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log (x^{9 + 27}) = 5$

Step 1.3.2

Add $9$ and $27$ .

$\log (x^{36}) = 5$

Step 2

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

$x \approx 1.37685716$

Step 3