Calculus Examples

$\log_{3} (y) + 3 \log_{3} (y^{2}) = 14$

Step 1

Simplify $\log_{3} (y) + 3 \log_{3} (y^{2})$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify $3 \log_{3} (y^{2})$ by moving $3$ inside the logarithm.

$\log_{3} (y) + \log_{3} ({(y^{2})}^{3}) = 14$

Step 1.1.2

Multiply the exponents in ${(y^{2})}^{3}$ .

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Step 1.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{3} (y) + \log_{3} (y^{2 \cdot 3}) = 14$

Step 1.1.2.2

Multiply $2$ by $3$ .

$\log_{3} (y) + \log_{3} (y^{6}) = 14$

Step 1.2

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

$\log_{3} (y \cdot y^{6}) = 14$

Step 1.3

Multiply $y$ by $y^{6}$ by adding the exponents.

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Step 1.3.1

Multiply $y$ by $y^{6}$ .

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Step 1.3.1.1

Raise $y$ to the power of $1$ .

$\log_{3} (y^{1} y^{6}) = 14$

Step 1.3.1.2

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

$\log_{3} (y^{1 + 6}) = 14$

Step 1.3.2

Add $1$ and $6$ .

$\log_{3} (y^{7}) = 14$

Step 2

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

$y = 9$

Step 3