Precalculus Examples

$\frac{1}{4} \cdot \log_{5} (81) - (2 \log_{5} (6) - \frac{1}{2} \cdot \log_{5} (4))$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify $\frac{1}{4} \log_{5} (81)$ by moving $\frac{1}{4}$ inside the logarithm.

$\log_{5} (81^{\frac{1}{4}}) - (2 \log_{5} (6) - \frac{1}{2} \cdot \log_{5} (4))$

Step 1.2

Rewrite $81$ as $3^{4}$ .

$\log_{5} ({(3^{4})}^{\frac{1}{4}}) - (2 \log_{5} (6) - \frac{1}{2} \cdot \log_{5} (4))$

Step 1.3

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

$\log_{5} (3^{4 (\frac{1}{4})}) - (2 \log_{5} (6) - \frac{1}{2} \cdot \log_{5} (4))$

Step 1.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.4.1

Cancel the common factor.

$\log_{5} (3^{4 (\frac{1}{4})}) - (2 \log_{5} (6) - \frac{1}{2} \cdot \log_{5} (4))$

Step 1.4.2

Rewrite the expression.

$\log_{5} (3^{1}) - (2 \log_{5} (6) - \frac{1}{2} \cdot \log_{5} (4))$

Step 1.5

Evaluate the exponent.

$\log_{5} (3) - (2 \log_{5} (6) - \frac{1}{2} \cdot \log_{5} (4))$

Step 1.6

Simplify each term.

Tap for more steps...

Step 1.6.1

Simplify $2 \log_{5} (6)$ by moving $2$ inside the logarithm.

$\log_{5} (3) - (\log_{5} (6^{2}) - \frac{1}{2} \cdot \log_{5} (4))$

Step 1.6.2

Raise $6$ to the power of $2$ .

$\log_{5} (3) - (\log_{5} (36) - \frac{1}{2} \cdot \log_{5} (4))$

Step 1.6.3

Multiply $- \frac{1}{2} \log_{5} (4)$ .

Tap for more steps...

Step 1.6.3.1

Reorder $\log_{5} (4)$ and $\frac{1}{2}$ .

$\log_{5} (3) - (\log_{5} (36) - (\frac{1}{2} \log_{5} (4)))$

Step 1.6.3.2

Simplify $\frac{1}{2} \log_{5} (4)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{5} (3) - (\log_{5} (36) - \log_{5} (4^{\frac{1}{2}}))$

Step 1.6.4

Rewrite $4$ as $2^{2}$ .

$\log_{5} (3) - (\log_{5} (36) - \log_{5} ({(2^{2})}^{\frac{1}{2}}))$

Step 1.6.5

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

$\log_{5} (3) - (\log_{5} (36) - \log_{5} (2^{2 (\frac{1}{2})}))$

Step 1.6.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.6.6.1

Cancel the common factor.

$\log_{5} (3) - (\log_{5} (36) - \log_{5} (2^{2 (\frac{1}{2})}))$

Step 1.6.6.2

Rewrite the expression.

$\log_{5} (3) - (\log_{5} (36) - \log_{5} (2^{1}))$

Step 1.6.7

Evaluate the exponent.

$\log_{5} (3) - (\log_{5} (36) - \log_{5} (2))$

Step 1.7

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{5} (3) - \log_{5} (\frac{36}{2})$

Step 1.8

Divide $36$ by $2$ .

$\log_{5} (3) - \log_{5} (18)$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{5} (\frac{3}{18})$

Step 3

Cancel the common factor of $3$ and $18$ .

Tap for more steps...

Step 3.1

Factor $3$ out of $3$ .

$\log_{5} (\frac{3 (1)}{18})$

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

Factor $3$ out of $18$ .

$\log_{5} (\frac{3 \cdot 1}{3 \cdot 6})$

Step 3.2.2

Cancel the common factor.

$\log_{5} (\frac{3 \cdot 1}{3 \cdot 6})$

Step 3.2.3

Rewrite the expression.

$\log_{5} (\frac{1}{6})$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\log_{5} (\frac{1}{6})$

Decimal Form:

$- 1.11328275 \dots$