Precalculus Examples

$\log (32) - \log_{16} (\frac{1}{2})$

Step 1

Simplify each term.

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Step 1.1

Logarithm base $16$ of $\frac{1}{2}$ is $- \frac{1}{4}$ .

$\log (32) - - \frac{1}{4}$

Step 1.2

Multiply $- - \frac{1}{4}$ .

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Step 1.2.1

Multiply $- 1$ by $- 1$ .

$\log (32) + 1 (\frac{1}{4})$

Step 1.2.2

Multiply $\frac{1}{4}$ by $1$ .

$\log (32) + \frac{1}{4}$

Step 2

To write $\log (32)$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\log (32) \cdot \frac{4}{4} + \frac{1}{4}$

Step 3

Combine fractions.

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Step 3.1

Combine $\log (32)$ and $\frac{4}{4}$ .

$\frac{\log (32) \cdot 4}{4} + \frac{1}{4}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{\log (32) \cdot 4 + 1}{4}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $\log (32) \cdot 4$ .

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Step 4.1.1

Reorder $\log (32)$ and $4$ .

$\frac{4 \cdot \log (32) + 1}{4}$

Step 4.1.2

Simplify $4 \cdot \log (32)$ by moving $4$ inside the logarithm.

$\frac{\log (32^{4}) + 1}{4}$

Step 4.2

Raise $32$ to the power of $4$ .

$\frac{\log (1048576) + 1}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (1048576) + 1}{4}$

Decimal Form:

$1.75514997 \dots$