Precalculus Examples

$\log (56) + \frac{1}{2} \cdot \log (7) - \log (8)$

Step 1

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

$\frac{1}{2} \cdot \log (7) + \log (\frac{56}{8})$

Step 2

Simplify each term.

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Step 2.1

Simplify $\frac{1}{2} \log (7)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log (7^{\frac{1}{2}}) + \log (\frac{56}{8})$

Step 2.2

Divide $56$ by $8$ .

$\log (7^{\frac{1}{2}}) + \log (7)$

Step 3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (7^{\frac{1}{2}} \cdot 7)$

Step 4

Multiply $7^{\frac{1}{2}}$ by $7$ by adding the exponents.

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Step 4.1

Multiply $7^{\frac{1}{2}}$ by $7$ .

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Step 4.1.1

Raise $7$ to the power of $1$ .

$\log (7^{\frac{1}{2}} \cdot 7^{1})$

Step 4.1.2

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

$\log (7^{\frac{1}{2} + 1})$

Step 4.2

Write $1$ as a fraction with a common denominator.

$\log (7^{\frac{1}{2} + \frac{2}{2}})$

Step 4.3

Combine the numerators over the common denominator.

$\log (7^{\frac{1 + 2}{2}})$

Step 4.4

Add $1$ and $2$ .

$\log (7^{\frac{3}{2}})$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\log (7^{\frac{3}{2}})$

Decimal Form:

$1.26764706 \dots$