Precalculus Examples

$\frac{2 \log (5) + 2 \log (7)}{4 \log (5) - \log (7)}$

Step 1

Simplify the numerator.

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Step 1.1

Simplify $2 \log (5)$ by moving $2$ inside the logarithm.

$\frac{\log (5^{2}) + 2 \log (7)}{4 \log (5) - \log (7)}$

Step 1.2

Raise $5$ to the power of $2$ .

$\frac{\log (25) + 2 \log (7)}{4 \log (5) - \log (7)}$

Step 1.3

Simplify $2 \log (7)$ by moving $2$ inside the logarithm.

$\frac{\log (25) + \log (7^{2})}{4 \log (5) - \log (7)}$

Step 1.4

Raise $7$ to the power of $2$ .

$\frac{\log (25) + \log (49)}{4 \log (5) - \log (7)}$

Step 1.5

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

$\frac{\log (25 \cdot 49)}{4 \log (5) - \log (7)}$

Step 1.6

Multiply $25$ by $49$ .

$\frac{\log (1225)}{4 \log (5) - \log (7)}$

Step 2

Simplify the denominator.

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Step 2.1

Simplify $4 \log (5)$ by moving $4$ inside the logarithm.

$\frac{\log (1225)}{\log (5^{4}) - \log (7)}$

Step 2.2

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

$\frac{\log (1225)}{\log (625) - \log (7)}$

Step 2.3

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

$\frac{\log (1225)}{\log (\frac{625}{7})}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (1225)}{\log (\frac{625}{7})}$

Decimal Form:

$1.58302471 \dots$