Trigonometry Examples

$\log_{64} (4) = w$

Step 1

Rewrite the equation as $w = \log_{64} (4)$ .

$w = \log_{64} (4)$

Step 2

Logarithm base $64$ of $4$ is $\frac{1}{3}$ .

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Step 2.1

Rewrite as an equation.

$w = \log_{64} (4) = x$

Step 2.2

Rewrite $\log_{64} (4) = x$ 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 $b^{y} = x$ .

$w = 64^{x} = 4$

Step 2.3

Create expressions in the equation that all have equal bases.

$w = {(2^{6})}^{x} = 2^{2}$

Step 2.4

Rewrite ${(2^{6})}^{x}$ as $2^{6 x}$ .

$w = 2^{6 x} = 2^{2}$

Step 2.5

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

$w = 6 x = 2$

Step 2.6

Solve for $x$ .

$w = x = \frac{1}{3}$

Step 2.7

The variable $x$ is equal to $\frac{1}{3}$ .

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Step 2.7.1

Factor $2$ out of $2$ .

$w = \frac{2 (1)}{6}$

Step 2.7.2

Cancel the common factors.

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Step 2.7.2.1

Factor $2$ out of $6$ .

$w = \frac{2 \cdot 1}{2 \cdot 3}$

Step 2.7.2.2

Cancel the common factor.

$w = \frac{2 \cdot 1}{2 \cdot 3}$

Step 2.7.2.3

Rewrite the expression.

$w = \frac{1}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$w = \frac{1}{3}$

Decimal Form:

$w = 0.\overline{3}$