Trigonometry Examples

$2 \log_{5} (x) - \log_{5} (5) = \log_{5} (125)$

Step 1

Simplify each term.

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Step 1.1

Logarithm base $5$ of $5$ is $1$ .

$2 \log_{5} (x) - 1 \cdot 1 = \log_{5} (125)$

Step 1.2

Multiply $- 1$ by $1$ .

$2 \log_{5} (x) - 1 = \log_{5} (125)$

Step 2

Logarithm base $5$ of $125$ is $3$ .

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Step 2.1

Rewrite as an equation.

$\log_{5} (125) = x$

Step 2.2

Rewrite $\log_{5} (125) = x$ 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 $b^{y} = x$ .

$5^{x} = 125$

Step 2.3

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

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

Step 2.4

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

$x = 3$

Step 2.5

The variable $x$ is equal to $3$ .

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

Step 3

Move all terms not containing $\log_{5} (x)$ to the right side of the equation.

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Step 3.1

Add $1$ to both sides of the equation.

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

Step 3.2

Add $3$ and $1$ .

$2 \log_{5} (x) = 4$

Step 4

Divide each term in $2 \log_{5} (x) = 4$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 \log_{5} (x) = 4$ by $2$ .

$\frac{2 \log_{5} (x)}{2} = \frac{4}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 \log_{5} (x)}{2} = \frac{4}{2}$

Step 4.2.1.2

Divide $\log_{5} (x)$ by $1$ .

$\log_{5} (x) = \frac{4}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $4$ by $2$ .

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

Step 5

Rewrite $\log_{5} (x) = 2$ 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 $b^{y} = x$ .

$5^{2} = x$

Step 6

Solve for $x$ .

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Step 6.1

Rewrite the equation as $x = 5^{2}$ .

$x = 5^{2}$

Step 6.2

Raise $5$ to the power of $2$ .

$x = 25$