Trigonometry Examples

$\log_{125} (25) = x$

Step 1

Logarithm base $125$ of $25$ is $\frac{2}{3}$ .

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Step 1.1

Rewrite as an equation.

$\log_{125} (25) = x = x$

Step 1.2

Rewrite $\log_{125} (25) = x$ 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 $b^{y} = x$ .

$125^{x} = 25 = x$

Step 1.3

Create expressions in the equation that all have equal bases.

${(5^{3})}^{x} = 5^{2} = x$

Step 1.4

Rewrite ${(5^{3})}^{x}$ as $5^{3 x}$ .

$5^{3 x} = 5^{2} = x$

Step 1.5

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

$3 x = 2 = x$

Step 1.6

Solve for $x$ .

$x = \frac{2}{3} = x$

Step 1.7

The variable $x$ is equal to $\frac{2}{3}$ .

$\frac{2}{3} = x$

Step 2

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

$x = \frac{2}{3}$

Step 3