Precalculus Examples

$2^{\log_{2} (5)} - 3 \log_{5} (\sqrt[3]{5})$

Step 1

Simplify each term.

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Step 1.1

Exponentiation and log are inverse functions.

$5 - 3 \log_{5} (\sqrt[3]{5})$

Step 1.2

Simplify $- 3 \log_{5} (\sqrt[3]{5})$ by moving $3$ inside the logarithm.

$5 - \log_{5} ({\sqrt[3]{5}}^{3})$

Step 1.3

Rewrite ${\sqrt[3]{5}}^{3}$ as $5$ .

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Step 1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$5 - \log_{5} ({(5^{\frac{1}{3}})}^{3})$

Step 1.3.2

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

$5 - \log_{5} (5^{\frac{1}{3} \cdot 3})$

Step 1.3.3

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

$5 - \log_{5} (5^{\frac{3}{3}})$

Step 1.3.4

Cancel the common factor of $3$ .

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Step 1.3.4.1

Cancel the common factor.

$5 - \log_{5} (5^{\frac{3}{3}})$

Step 1.3.4.2

Rewrite the expression.

$5 - \log_{5} (5^{1})$

Step 1.3.5

Evaluate the exponent.

$5 - \log_{5} (5)$

Step 1.4

Logarithm base $5$ of $5$ is $1$ .

$5 - 1 \cdot 1$

Step 1.5

Multiply $- 1$ by $1$ .

$5 - 1$

Step 2

Subtract $1$ from $5$ .

$4$