Precalculus Examples

$4 \log_{4} (11) - 3 \log_{7} (\sqrt[3]{7})$

Step 1

Simplify $4 \log_{4} (11)$ by moving $4$ inside the logarithm.

$\log_{4} (11^{4}) - 3 \log_{7} (\sqrt[3]{7})$

Step 2

Raise $11$ to the power of $4$ .

$\log_{4} (14641) - 3 \log_{7} (\sqrt[3]{7})$

Step 3

Simplify $- 3 \log_{7} (\sqrt[3]{7})$ by moving $3$ inside the logarithm.

$\log_{4} (14641) - \log_{7} ({\sqrt[3]{7}}^{3})$

Step 4

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$\log_{4} (14641) - \log_{7} ({(7^{\frac{1}{3}})}^{3})$

Step 4.2

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

$\log_{4} (14641) - \log_{7} (7^{\frac{1}{3} \cdot 3})$

Step 4.3

Combine $\frac{1}{3}$ and $3$ .

$\log_{4} (14641) - \log_{7} (7^{\frac{3}{3}})$

Step 4.4

Cancel the common factor of $3$ .

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Step 4.4.1

Cancel the common factor.

$\log_{4} (14641) - \log_{7} (7^{\frac{3}{3}})$

Step 4.4.2

Rewrite the expression.

$\log_{4} (14641) - \log_{7} (7^{1})$

Step 4.5

Evaluate the exponent.

$\log_{4} (14641) - \log_{7} (7)$

Step 5

Logarithm base $7$ of $7$ is $1$ .

$\log_{4} (14641) - 1 \cdot 1$

Step 6

Multiply $- 1$ by $1$ .

$\log_{4} (14641) - 1$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\log_{4} (14641) - 1$

Decimal Form:

$5.91886323 \dots$