Algebra Examples

$\frac{3}{2} \ln (16 x^{8}) - \frac{1}{5} \ln (2 y^{25})$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify $\frac{3}{2} \ln (16 x^{8})$ by moving $\frac{3}{2}$ inside the logarithm.

$\ln ({(16 x^{8})}^{\frac{3}{2}}) - \frac{1}{5} \cdot \ln (2 y^{25})$

Step 1.2

Apply the product rule to $16 x^{8}$ .

$\ln (16^{\frac{3}{2}} {(x^{8})}^{\frac{3}{2}}) - \frac{1}{5} \cdot \ln (2 y^{25})$

Step 1.3

Rewrite $16$ as $4^{2}$ .

$\ln ({(4^{2})}^{\frac{3}{2}} {(x^{8})}^{\frac{3}{2}}) - \frac{1}{5} \cdot \ln (2 y^{25})$

Step 1.4

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

$\ln (4^{2 (\frac{3}{2})} {(x^{8})}^{\frac{3}{2}}) - \frac{1}{5} \cdot \ln (2 y^{25})$

Step 1.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.5.1

Cancel the common factor.

$\ln (4^{2 (\frac{3}{2})} {(x^{8})}^{\frac{3}{2}}) - \frac{1}{5} \cdot \ln (2 y^{25})$

Step 1.5.2

Rewrite the expression.

$\ln (4^{3} {(x^{8})}^{\frac{3}{2}}) - \frac{1}{5} \cdot \ln (2 y^{25})$

Step 1.6

Raise $4$ to the power of $3$ .

$\ln (64 {(x^{8})}^{\frac{3}{2}}) - \frac{1}{5} \cdot \ln (2 y^{25})$

Step 1.7

Multiply the exponents in ${(x^{8})}^{\frac{3}{2}}$ .

Tap for more steps...

Step 1.7.1

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

$\ln (64 x^{8 (\frac{3}{2})}) - \frac{1}{5} \cdot \ln (2 y^{25})$

Step 1.7.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.7.2.1

Factor $2$ out of $8$ .

$\ln (64 x^{2 (4) \frac{3}{2}}) - \frac{1}{5} \cdot \ln (2 y^{25})$

Step 1.7.2.2

Cancel the common factor.

$\ln (64 x^{2 \cdot 4 \frac{3}{2}}) - \frac{1}{5} \cdot \ln (2 y^{25})$

Step 1.7.2.3

Rewrite the expression.

$\ln (64 x^{4 \cdot 3}) - \frac{1}{5} \cdot \ln (2 y^{25})$

Step 1.7.3

Multiply $4$ by $3$ .

$\ln (64 x^{12}) - \frac{1}{5} \cdot \ln (2 y^{25})$

Step 1.8

Multiply $- \frac{1}{5} \ln (2 y^{25})$ .

Tap for more steps...

Step 1.8.1

Reorder $\ln (2 y^{25})$ and $\frac{1}{5}$ .

$\ln (64 x^{12}) - (\frac{1}{5} \ln (2 y^{25}))$

Step 1.8.2

Simplify $\frac{1}{5} \ln (2 y^{25})$ by moving $\frac{1}{5}$ inside the logarithm.

$\ln (64 x^{12}) - \ln ({(2 y^{25})}^{\frac{1}{5}})$

Step 1.9

Apply the product rule to $2 y^{25}$ .

$\ln (64 x^{12}) - \ln (2^{\frac{1}{5}} {(y^{25})}^{\frac{1}{5}})$

Step 1.10

Multiply the exponents in ${(y^{25})}^{\frac{1}{5}}$ .

Tap for more steps...

Step 1.10.1

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

$\ln (64 x^{12}) - \ln (2^{\frac{1}{5}} y^{25 (\frac{1}{5})})$

Step 1.10.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 1.10.2.1

Factor $5$ out of $25$ .

$\ln (64 x^{12}) - \ln (2^{\frac{1}{5}} y^{5 (5) \frac{1}{5}})$

Step 1.10.2.2

Cancel the common factor.

$\ln (64 x^{12}) - \ln (2^{\frac{1}{5}} y^{5 \cdot 5 \frac{1}{5}})$

Step 1.10.2.3

Rewrite the expression.

$\ln (64 x^{12}) - \ln (2^{\frac{1}{5}} y^{5})$

Step 2

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

$\ln (\frac{64 x^{12}}{2^{\frac{1}{5}} y^{5}})$

Step 3

Reorder factors in $\ln (\frac{64 x^{12}}{2^{\frac{1}{5}} y^{5}})$ .

$\ln (\frac{64 x^{12}}{y^{5} \cdot 2^{\frac{1}{5}}})$