Precalculus Examples

$\log_{2} (56) - \log_{4} (49)$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\log_{2} (56)$ using the change of base formula.

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Step 1.1.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} - \log_{4} (49)$

Step 1.1.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (56)}{\log (2)} - \log_{4} (49)$

Step 1.2

Rewrite $\log_{4} (49)$ using the change of base formula.

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Step 1.2.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\log (56)}{\log (2)} - \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 1.2.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (56)}{\log (2)} - \frac{\log (49)}{\log (4)}$

Step 1.3

Rewrite $\log (49)$ as $\log (7^{2})$ .

$\frac{\log (56)}{\log (2)} - \frac{\log (7^{2})}{\log (4)}$

Step 1.4

Expand $\log (7^{2})$ by moving $2$ outside the logarithm.

$\frac{\log (56)}{\log (2)} - \frac{2 \log (7)}{\log (4)}$

Step 1.5

Rewrite $\log (4)$ as $\log (2^{2})$ .

$\frac{\log (56)}{\log (2)} - \frac{2 \log (7)}{\log (2^{2})}$

Step 1.6

Expand $\log (2^{2})$ by moving $2$ outside the logarithm.

$\frac{\log (56)}{\log (2)} - \frac{2 \log (7)}{2 \log (2)}$

Step 1.7

Cancel the common factor of $2$ .

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Step 1.7.1

Cancel the common factor.

$\frac{\log (56)}{\log (2)} - \frac{2 \log (7)}{2 \log (2)}$

Step 1.7.2

Rewrite the expression.

$\frac{\log (56)}{\log (2)} - \frac{\log (7)}{\log (2)}$

Step 2

Combine the numerators over the common denominator.

$\frac{\log (56) - \log (7)}{\log (2)}$

Step 3

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

$\frac{\log (\frac{56}{7})}{\log (2)}$

Step 4

Divide $56$ by $7$ .

$\frac{\log (8)}{\log (2)}$

Step 5

Rewrite $\log (8)$ as $\log (2^{3})$ .

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

Step 6

Expand $\log (2^{3})$ by moving $3$ outside the logarithm.

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

Step 7

Cancel the common factor of $\log (2)$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Divide $3$ by $1$ .

$3$