Precalculus Examples

$\frac{1}{4} \cdot (\frac{\ln (93)}{4} - \frac{\ln (24)}{4})$

Step 1

Combine the numerators over the common denominator.

$\frac{1}{4} \cdot \frac{\ln (93) - \ln (24)}{4}$

Step 2

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

$\frac{1}{4} \cdot \frac{\ln (\frac{93}{24})}{4}$

Step 3

Cancel the common factor of $93$ and $24$ .

Tap for more steps...

Step 3.1

Factor $3$ out of $93$ .

$\frac{1}{4} \cdot \frac{\ln (\frac{3 (31)}{24})}{4}$

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

Factor $3$ out of $24$ .

$\frac{1}{4} \cdot \frac{\ln (\frac{3 \cdot 31}{3 \cdot 8})}{4}$

Step 3.2.2

Cancel the common factor.

$\frac{1}{4} \cdot \frac{\ln (\frac{3 \cdot 31}{3 \cdot 8})}{4}$

Step 3.2.3

Rewrite the expression.

$\frac{1}{4} \cdot \frac{\ln (\frac{31}{8})}{4}$

Step 4

Rewrite $\frac{\ln (\frac{31}{8})}{4}$ as $\frac{1}{4} \ln (\frac{31}{8})$ .

$\frac{1}{4} \cdot (\frac{1}{4} \ln (\frac{31}{8}))$

Step 5

Simplify $\frac{1}{4} \ln (\frac{31}{8})$ by moving $\frac{1}{4}$ inside the logarithm.

$\frac{1}{4} \cdot \ln ({(\frac{31}{8})}^{\frac{1}{4}})$

Step 6

Apply the product rule to $\frac{31}{8}$ .

$\frac{1}{4} \cdot \ln (\frac{31^{\frac{1}{4}}}{8^{\frac{1}{4}}})$

Step 7

Simplify $\frac{1}{4} \ln (\frac{31^{\frac{1}{4}}}{8^{\frac{1}{4}}})$ by moving $\frac{1}{4}$ inside the logarithm.

$\ln ({(\frac{31^{\frac{1}{4}}}{8^{\frac{1}{4}}})}^{\frac{1}{4}})$

Step 8

Apply the product rule to $\frac{31^{\frac{1}{4}}}{8^{\frac{1}{4}}}$ .

$\ln (\frac{{(31^{\frac{1}{4}})}^{\frac{1}{4}}}{{(8^{\frac{1}{4}})}^{\frac{1}{4}}})$

Step 9

Multiply the exponents in ${(31^{\frac{1}{4}})}^{\frac{1}{4}}$ .

Tap for more steps...

Step 9.1

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

$\ln (\frac{31^{\frac{1}{4} \cdot \frac{1}{4}}}{{(8^{\frac{1}{4}})}^{\frac{1}{4}}})$

Step 9.2

Multiply $\frac{1}{4} \cdot \frac{1}{4}$ .

Tap for more steps...

Step 9.2.1

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

$\ln (\frac{31^{\frac{1}{4 \cdot 4}}}{{(8^{\frac{1}{4}})}^{\frac{1}{4}}})$

Step 9.2.2

Multiply $4$ by $4$ .

$\ln (\frac{31^{\frac{1}{16}}}{{(8^{\frac{1}{4}})}^{\frac{1}{4}}})$

Step 10

Multiply the exponents in ${(8^{\frac{1}{4}})}^{\frac{1}{4}}$ .

Tap for more steps...

Step 10.1

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

$\ln (\frac{31^{\frac{1}{16}}}{8^{\frac{1}{4} \cdot \frac{1}{4}}})$

Step 10.2

Multiply $\frac{1}{4} \cdot \frac{1}{4}$ .

Tap for more steps...

Step 10.2.1

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

$\ln (\frac{31^{\frac{1}{16}}}{8^{\frac{1}{4 \cdot 4}}})$

Step 10.2.2

Multiply $4$ by $4$ .

$\ln (\frac{31^{\frac{1}{16}}}{8^{\frac{1}{16}}})$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{31^{\frac{1}{16}}}{8^{\frac{1}{16}}})$

Decimal Form:

$0.08465910 \dots$