Trigonometry Examples

$x = \log_{3} (81)$

Step 1

Logarithm base $3$ of $81$ is $4$ .

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Step 1.1

Rewrite as an equation.

$x = \log_{3} (81) = x$

Step 1.2

Rewrite $\log_{3} (81) = x$ 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 $b^{y} = x$ .

$x = 3^{x} = 81$

Step 1.3

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

$x = 3^{x} = 3^{4}$

Step 1.4

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

$x = x = 4$

Step 1.5

The variable $x$ is equal to $4$ .

$x = 4$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 4$

Step 3