Trigonometry Examples

$2 \log_{2} (x) + \log_{2} (\frac{1}{32}) = 3 \log_{2} (x) - 3 \log_{2} (2)$

Step 1

Logarithm base $2$ of $\frac{1}{32}$ is $- 5$ .

$2 \log_{2} (x) - 5 = 3 \log_{2} (x) - 3 \log_{2} (2)$

Step 2

Simplify each term.

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Step 2.1

Logarithm base $2$ of $2$ is $1$ .

$2 \log_{2} (x) - 5 = 3 \log_{2} (x) - 3 \cdot 1$

Step 2.2

Multiply $- 3$ by $1$ .

$2 \log_{2} (x) - 5 = 3 \log_{2} (x) - 3$

Step 3

Move all the terms containing a logarithm to the left side of the equation.

$2 \log_{2} (x) - 3 \log_{2} (x) = 5 - 3$

Step 4

Subtract $3 \log_{2} (x)$ from $2 \log_{2} (x)$ .

$- \log_{2} (x) = 5 - 3$

Step 5

Subtract $3$ from $5$ .

$- \log_{2} (x) = 2$

Step 6

Divide each term in $- \log_{2} (x) = 2$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- \log_{2} (x) = 2$ by $- 1$ .

$\frac{- \log_{2} (x)}{- 1} = \frac{2}{- 1}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{\log_{2} (x)}{1} = \frac{2}{- 1}$

Step 6.2.2

Divide $\log_{2} (x)$ by $1$ .

$\log_{2} (x) = \frac{2}{- 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $2$ by $- 1$ .

$\log_{2} (x) = - 2$

Step 7

Rewrite $\log_{2} (x) = - 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$2^{- 2} = x$

Step 8

Solve for $x$ .

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Step 8.1

Rewrite the equation as $x = 2^{- 2}$ .

$x = 2^{- 2}$

Step 8.2

Simplify $2^{- 2}$ .

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Step 8.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x = \frac{1}{2^{2}}$

Step 8.2.2

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

$x = \frac{1}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{4}$

Decimal Form:

$x = 0.25$