Precalculus Examples

$\frac{\log (b) (3^{3} \cdot 5^{4})}{2^{2}}$

Step 1

Simplify the numerator.

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

$\frac{\log (b) \cdot 27 \cdot 5^{4}}{2^{2}}$

Raise $5$ to the power of $4$ .

$\frac{\log (b) \cdot 27 \cdot 625}{2^{2}}$

Combine exponents.

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Reorder $\log (b)$ and $27$ .

$\frac{27 \cdot \log (b) \cdot 625}{2^{2}}$

Simplify $27 \cdot \log (b)$ by moving $27$ inside the logarithm.

$\frac{\log (b^{27}) \cdot 625}{2^{2}}$

Reorder $\log (b^{27})$ and $625$ .

$\frac{625 \cdot \log (b^{27})}{2^{2}}$

Simplify $625 \cdot \log (b^{27})$ by moving $625$ inside the logarithm.

$\frac{\log ({(b^{27})}^{625})}{2^{2}}$

Multiply the exponents in ${(b^{27})}^{625}$ .

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

$\frac{\log (b^{27 \cdot 625})}{2^{2}}$

Multiply $27$ by $625$ .

$\frac{\log (b^{16875})}{2^{2}}$

Step 2

Simplify the expression.

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

$\frac{\log (b^{16875})}{4}$

Rewrite $\frac{\log (b^{16875})}{4}$ as $\frac{1}{4} \log (b^{16875})$ .

$\frac{1}{4} \log (b^{16875})$

Step 3

Simplify $\frac{1}{4} \log (b^{16875})$ by moving $\frac{1}{4}$ inside the logarithm.

$\log ({(b^{16875})}^{\frac{1}{4}})$

Step 4

Multiply the exponents in ${(b^{16875})}^{\frac{1}{4}}$ .

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

$\log (b^{16875 (\frac{1}{4})})$

Combine $16875$ and $\frac{1}{4}$ .

$\log (b^{\frac{16875}{4}})$