Calculus Examples

$\log_{3} (x) + 13 \log_{3} (x^{4}) = 14$

Step 1

Simplify $\log_{3} (x) + 13 \log_{3} (x^{4})$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify $13 \log_{3} (x^{4})$ by moving $13$ inside the logarithm.

$\log_{3} (x) + \log_{3} ({(x^{4})}^{13}) = 14$

Step 1.1.2

Multiply the exponents in ${(x^{4})}^{13}$ .

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Step 1.1.2.1

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

$\log_{3} (x) + \log_{3} (x^{4 \cdot 13}) = 14$

Step 1.1.2.2

Multiply $4$ by $13$ .

$\log_{3} (x) + \log_{3} (x^{52}) = 14$

Step 1.2

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

$\log_{3} (x \cdot x^{52}) = 14$

Step 1.3

Multiply $x$ by $x^{52}$ by adding the exponents.

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Step 1.3.1

Multiply $x$ by $x^{52}$ .

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Step 1.3.1.1

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

$\log_{3} (x^{1} x^{52}) = 14$

Step 1.3.1.2

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

$\log_{3} (x^{1 + 52}) = 14$

Step 1.3.2

Add $1$ and $52$ .

$\log_{3} (x^{53}) = 14$

Step 2

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

$x \approx 1.33669409$

Step 3