Precalculus Examples

$\frac{3}{2} \cdot \ln (5 t^{6}) - \frac{3}{4} \cdot \ln (t^{4})$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{3}{2} \ln (5 t^{6})$ by moving $\frac{3}{2}$ inside the logarithm.

$\ln ({(5 t^{6})}^{\frac{3}{2}}) - \frac{3}{4} \cdot \ln (t^{4})$

Step 1.2

Apply the product rule to $5 t^{6}$ .

$\ln (5^{\frac{3}{2}} {(t^{6})}^{\frac{3}{2}}) - \frac{3}{4} \cdot \ln (t^{4})$

Step 1.3

Multiply the exponents in ${(t^{6})}^{\frac{3}{2}}$ .

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Step 1.3.1

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

$\ln (5^{\frac{3}{2}} t^{6 (\frac{3}{2})}) - \frac{3}{4} \cdot \ln (t^{4})$

Step 1.3.2

Cancel the common factor of $2$ .

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Step 1.3.2.1

Factor $2$ out of $6$ .

$\ln (5^{\frac{3}{2}} t^{2 (3) \frac{3}{2}}) - \frac{3}{4} \cdot \ln (t^{4})$

Step 1.3.2.2

Cancel the common factor.

$\ln (5^{\frac{3}{2}} t^{2 \cdot 3 \frac{3}{2}}) - \frac{3}{4} \cdot \ln (t^{4})$

Step 1.3.2.3

Rewrite the expression.

$\ln (5^{\frac{3}{2}} t^{3 \cdot 3}) - \frac{3}{4} \cdot \ln (t^{4})$

Step 1.3.3

Multiply $3$ by $3$ .

$\ln (5^{\frac{3}{2}} t^{9}) - \frac{3}{4} \cdot \ln (t^{4})$

Step 1.4

Multiply $- \frac{3}{4} \ln (t^{4})$ .

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Step 1.4.1

Reorder $\ln (t^{4})$ and $\frac{3}{4}$ .

$\ln (5^{\frac{3}{2}} t^{9}) - (\frac{3}{4} \ln (t^{4}))$

Step 1.4.2

Simplify $\frac{3}{4} \ln (t^{4})$ by moving $\frac{3}{4}$ inside the logarithm.

$\ln (5^{\frac{3}{2}} t^{9}) - \ln ({(t^{4})}^{\frac{3}{4}})$

Step 1.5

Multiply the exponents in ${(t^{4})}^{\frac{3}{4}}$ .

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Step 1.5.1

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

$\ln (5^{\frac{3}{2}} t^{9}) - \ln (t^{4 (\frac{3}{4})})$

Step 1.5.2

Cancel the common factor of $4$ .

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Step 1.5.2.1

Cancel the common factor.

$\ln (5^{\frac{3}{2}} t^{9}) - \ln (t^{4 (\frac{3}{4})})$

Step 1.5.2.2

Rewrite the expression.

$\ln (5^{\frac{3}{2}} t^{9}) - \ln (t^{3})$

Step 2

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

$\ln (\frac{5^{\frac{3}{2}} t^{9}}{t^{3}})$

Step 3

Factor $t^{3}$ out of $5^{\frac{3}{2}} t^{9}$ .

$\ln (\frac{t^{3} (5^{\frac{3}{2}} t^{6})}{t^{3}})$

Step 4

Cancel the common factors.

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Step 4.1

Multiply by $1$ .

$\ln (\frac{t^{3} (5^{\frac{3}{2}} t^{6})}{t^{3} \cdot 1})$

Step 4.2

Cancel the common factor.

$\ln (\frac{t^{3} (5^{\frac{3}{2}} t^{6})}{t^{3} \cdot 1})$

Step 4.3

Rewrite the expression.

$\ln (\frac{5^{\frac{3}{2}} t^{6}}{1})$

Step 4.4

Divide $5^{\frac{3}{2}} t^{6}$ by $1$ .

$\ln (5^{\frac{3}{2}} t^{6})$

Step 5

Reorder factors in $\ln (5^{\frac{3}{2}} t^{6})$ .

$\ln (t^{6} \cdot 5^{\frac{3}{2}})$