Expand the Logarithmic Expression log of b square root of 57/74

$\log (b) \sqrt{\frac{57}{74}}$

Step 1

Rewrite $\sqrt{\frac{57}{74}}$ as $\frac{\sqrt{57}}{\sqrt{74}}$ .

$\log (b) \frac{\sqrt{57}}{\sqrt{74}}$

Step 2

Multiply $\frac{\sqrt{57}}{\sqrt{74}}$ by $\frac{\sqrt{74}}{\sqrt{74}}$ .

$\log (b) (\frac{\sqrt{57}}{\sqrt{74}} \cdot \frac{\sqrt{74}}{\sqrt{74}})$

Step 3

Combine and simplify the denominator.

Multiply $\frac{\sqrt{57}}{\sqrt{74}}$ by $\frac{\sqrt{74}}{\sqrt{74}}$ .

$\log (b) \frac{\sqrt{57} \sqrt{74}}{\sqrt{74} \sqrt{74}}$

Raise $\sqrt{74}$ to the power of $1$ .

$\log (b) \frac{\sqrt{57} \sqrt{74}}{{\sqrt{74}}^{1} \sqrt{74}}$

Raise $\sqrt{74}$ to the power of $1$ .

$\log (b) \frac{\sqrt{57} \sqrt{74}}{{\sqrt{74}}^{1} {\sqrt{74}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log (b) \frac{\sqrt{57} \sqrt{74}}{{\sqrt{74}}^{1 + 1}}$

Add $1$ and $1$ .

$\log (b) \frac{\sqrt{57} \sqrt{74}}{{\sqrt{74}}^{2}}$

Rewrite ${\sqrt{74}}^{2}$ as $74$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{74}$ as $74^{\frac{1}{2}}$ .

$\log (b) \frac{\sqrt{57} \sqrt{74}}{{(74^{\frac{1}{2}})}^{2}}$

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

$\log (b) \frac{\sqrt{57} \sqrt{74}}{74^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\log (b) \frac{\sqrt{57} \sqrt{74}}{74^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\log (b) \frac{\sqrt{57} \sqrt{74}}{74^{\frac{2}{2}}}$

Rewrite the expression.

$\log (b) \frac{\sqrt{57} \sqrt{74}}{74^{1}}$

Evaluate the exponent.

$\log (b) \frac{\sqrt{57} \sqrt{74}}{74}$

Step 4

Simplify the numerator.

Combine using the product rule for radicals.

$\log (b) \frac{\sqrt{57 \cdot 74}}{74}$

Multiply $57$ by $74$ .

$\log (b) \frac{\sqrt{4218}}{74}$

Step 5

Combine $\log (b)$ and $\frac{\sqrt{4218}}{74}$ .

$\frac{\log (b) \sqrt{4218}}{74}$