Precalculus Examples

$\log_{2} (\cos (\frac{5 π}{3}))$

Step 1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\log_{2} (\cos (\frac{π}{3}))$

Step 2

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\log_{2} (\frac{1}{2})$

Step 3

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

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Step 3.1

Rewrite as an equation.

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

Step 3.2

Rewrite $\log_{2} (\frac{1}{2}) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

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

Step 3.3

Create equivalent expressions in the equation that all have equal bases.

$2^{x} = 2^{- 1}$

Step 3.4

Since the bases are the same, the two expressions are only equal if the exponents are also equal.

$x = - 1$

Step 3.5

The variable $x$ is equal to $- 1$ .

$- 1$