Precalculus Examples

$2 \log_{5} (18 - \log_{5} (3)) + \frac{1}{2} \cdot \log_{5} (16)$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{5} (18 - \log_{5} (3))$ by moving $2$ inside the logarithm.

$\log_{5} ({(18 - \log_{5} (3))}^{2}) + \frac{1}{2} \cdot \log_{5} (16)$

Step 1.2

Simplify $\frac{1}{2} \log_{5} (16)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{5} ({(18 - \log_{5} (3))}^{2}) + \log_{5} (16^{\frac{1}{2}})$

Step 1.3

Rewrite $16$ as $4^{2}$ .

$\log_{5} ({(18 - \log_{5} (3))}^{2}) + \log_{5} ({(4^{2})}^{\frac{1}{2}})$

Step 1.4

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

$\log_{5} ({(18 - \log_{5} (3))}^{2}) + \log_{5} (4^{2 (\frac{1}{2})})$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Cancel the common factor.

$\log_{5} ({(18 - \log_{5} (3))}^{2}) + \log_{5} (4^{2 (\frac{1}{2})})$

Step 1.5.2

Rewrite the expression.

$\log_{5} ({(18 - \log_{5} (3))}^{2}) + \log_{5} (4^{1})$

Step 1.6

Evaluate the exponent.

$\log_{5} ({(18 - \log_{5} (3))}^{2}) + \log_{5} (4)$

Step 2

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

$\log_{5} ({(18 - \log_{5} (3))}^{2} \cdot 4)$

Step 3

Move $4$ to the left of ${(18 - \log_{5} (3))}^{2}$ .

$\log_{5} (4 {(18 - \log_{5} (3))}^{2})$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\log_{5} (4 {(18 - \log_{5} (3))}^{2})$

Decimal Form:

$4.40508897 \dots$