Precalculus Examples

$\frac{3}{2} \cdot \ln ({(16 x)}^{10}) - \frac{2}{5} \cdot \ln ({(2 y)}^{30})$

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $16 x$ .

$\frac{3}{2} \cdot \ln (16^{10} x^{10}) - \frac{2}{5} \cdot \ln ({(2 y)}^{30})$

Step 1.2

Raise $16$ to the power of $10$ .

$\frac{3}{2} \cdot \ln (1099511627776 x^{10}) - \frac{2}{5} \cdot \ln ({(2 y)}^{30})$

Step 1.3

Simplify $\frac{3}{2} \ln (1099511627776 x^{10})$ by moving $\frac{3}{2}$ inside the logarithm.

$\ln ({(1099511627776 x^{10})}^{\frac{3}{2}}) - \frac{2}{5} \cdot \ln ({(2 y)}^{30})$

Step 1.4

Apply the product rule to $1099511627776 x^{10}$ .

$\ln (1099511627776^{\frac{3}{2}} {(x^{10})}^{\frac{3}{2}}) - \frac{2}{5} \cdot \ln ({(2 y)}^{30})$

Step 1.5

Rewrite $1099511627776$ as $1048576^{2}$ .

$\ln ({(1048576^{2})}^{\frac{3}{2}} {(x^{10})}^{\frac{3}{2}}) - \frac{2}{5} \cdot \ln ({(2 y)}^{30})$

Step 1.6

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

$\ln (1048576^{2 (\frac{3}{2})} {(x^{10})}^{\frac{3}{2}}) - \frac{2}{5} \cdot \ln ({(2 y)}^{30})$

Step 1.7

Cancel the common factor of $2$ .

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Step 1.7.1

Cancel the common factor.

$\ln (1048576^{2 (\frac{3}{2})} {(x^{10})}^{\frac{3}{2}}) - \frac{2}{5} \cdot \ln ({(2 y)}^{30})$

Step 1.7.2

Rewrite the expression.

$\ln (1048576^{3} {(x^{10})}^{\frac{3}{2}}) - \frac{2}{5} \cdot \ln ({(2 y)}^{30})$

Step 1.8

Raise $1048576$ to the power of $3$ .

$\ln (1152921504606850000 {(x^{10})}^{\frac{3}{2}}) - \frac{2}{5} \cdot \ln ({(2 y)}^{30})$

Step 1.9

Multiply the exponents in ${(x^{10})}^{\frac{3}{2}}$ .

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Step 1.9.1

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

$\ln (1152921504606850000 x^{10 (\frac{3}{2})}) - \frac{2}{5} \cdot \ln ({(2 y)}^{30})$

Step 1.9.2

Cancel the common factor of $2$ .

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Step 1.9.2.1

Factor $2$ out of $10$ .

$\ln (1152921504606850000 x^{2 (5) \frac{3}{2}}) - \frac{2}{5} \cdot \ln ({(2 y)}^{30})$

Step 1.9.2.2

Cancel the common factor.

$\ln (1152921504606850000 x^{2 \cdot 5 \frac{3}{2}}) - \frac{2}{5} \cdot \ln ({(2 y)}^{30})$

Step 1.9.2.3

Rewrite the expression.

$\ln (1152921504606850000 x^{5 \cdot 3}) - \frac{2}{5} \cdot \ln ({(2 y)}^{30})$

Step 1.9.3

Multiply $5$ by $3$ .

$\ln (1152921504606850000 x^{15}) - \frac{2}{5} \cdot \ln ({(2 y)}^{30})$

Step 1.10

Apply the product rule to $2 y$ .

$\ln (1152921504606850000 x^{15}) - \frac{2}{5} \cdot \ln (2^{30} y^{30})$

Step 1.11

Raise $2$ to the power of $30$ .

$\ln (1152921504606850000 x^{15}) - \frac{2}{5} \cdot \ln (1073741824 y^{30})$

Step 1.12

Multiply $- \frac{2}{5} \ln (1073741824 y^{30})$ .

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Step 1.12.1

Reorder $\ln (1073741824 y^{30})$ and $\frac{2}{5}$ .

$\ln (1152921504606850000 x^{15}) - (\frac{2}{5} \ln (1073741824 y^{30}))$

Step 1.12.2

Simplify $\frac{2}{5} \ln (1073741824 y^{30})$ by moving $\frac{2}{5}$ inside the logarithm.

$\ln (1152921504606850000 x^{15}) - \ln ({(1073741824 y^{30})}^{\frac{2}{5}})$

Step 1.13

Apply the product rule to $1073741824 y^{30}$ .

$\ln (1152921504606850000 x^{15}) - \ln (1073741824^{\frac{2}{5}} {(y^{30})}^{\frac{2}{5}})$

Step 1.14

Rewrite $1073741824$ as $64^{5}$ .

$\ln (1152921504606850000 x^{15}) - \ln ({(64^{5})}^{\frac{2}{5}} {(y^{30})}^{\frac{2}{5}})$

Step 1.15

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

$\ln (1152921504606850000 x^{15}) - \ln (64^{5 (\frac{2}{5})} {(y^{30})}^{\frac{2}{5}})$

Step 1.16

Cancel the common factor of $5$ .

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Step 1.16.1

Cancel the common factor.

$\ln (1152921504606850000 x^{15}) - \ln (64^{5 (\frac{2}{5})} {(y^{30})}^{\frac{2}{5}})$

Step 1.16.2

Rewrite the expression.

$\ln (1152921504606850000 x^{15}) - \ln (64^{2} {(y^{30})}^{\frac{2}{5}})$

Step 1.17

Raise $64$ to the power of $2$ .

$\ln (1152921504606850000 x^{15}) - \ln (4096 {(y^{30})}^{\frac{2}{5}})$

Step 1.18

Multiply the exponents in ${(y^{30})}^{\frac{2}{5}}$ .

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Step 1.18.1

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

$\ln (1152921504606850000 x^{15}) - \ln (4096 y^{30 (\frac{2}{5})})$

Step 1.18.2

Cancel the common factor of $5$ .

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Step 1.18.2.1

Factor $5$ out of $30$ .

$\ln (1152921504606850000 x^{15}) - \ln (4096 y^{5 (6) \frac{2}{5}})$

Step 1.18.2.2

Cancel the common factor.

$\ln (1152921504606850000 x^{15}) - \ln (4096 y^{5 \cdot 6 \frac{2}{5}})$

Step 1.18.2.3

Rewrite the expression.

$\ln (1152921504606850000 x^{15}) - \ln (4096 y^{6 \cdot 2})$

Step 1.18.3

Multiply $6$ by $2$ .

$\ln (1152921504606850000 x^{15}) - \ln (4096 y^{12})$

Step 2

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

$\ln (\frac{1152921504606850000 x^{15}}{4096 y^{12}})$

Step 3

Cancel the common factor of $1152921504606850000$ and $4096$ .

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Step 3.1

Factor $16$ out of $1152921504606850000 x^{15}$ .

$\ln (\frac{16 (72057594037928125 x^{15})}{4096 y^{12}})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $16$ out of $4096 y^{12}$ .

$\ln (\frac{16 (72057594037928125 x^{15})}{16 (256 y^{12})})$

Step 3.2.2

Cancel the common factor.

$\ln (\frac{16 (72057594037928125 x^{15})}{16 (256 y^{12})})$

Step 3.2.3

Rewrite the expression.

$\ln (\frac{72057594037928125 x^{15}}{256 y^{12}})$