Algebra Examples

$\log_{9} (5 x - 1) = \frac{1}{2}$

Step 1

Rewrite $\log_{9} (5 x - 1) = \frac{1}{2}$ 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 $b^{y} = x$ .

$9^{\frac{1}{2}} = 5 x - 1$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $5 x - 1 = 9^{\frac{1}{2}}$ .

$5 x - 1 = 9^{\frac{1}{2}}$

Step 2.2

Simplify $9^{\frac{1}{2}}$ .

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Step 2.2.1

Rewrite.

$5 x - 1 = 0 + 0 + 9^{\frac{1}{2}}$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Rewrite $9$ as $3^{2}$ .

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

Step 2.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.3

Cancel the common factor of $2$ .

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Step 2.2.3.1

Cancel the common factor.

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

Step 2.2.3.2

Rewrite the expression.

$5 x - 1 = 3^{1}$

Step 2.2.4

Evaluate the exponent.

$5 x - 1 = 3$

Step 2.3

Move all terms not containing $x$ to the right side of the equation.

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Step 2.3.1

Add $1$ to both sides of the equation.

$5 x = 3 + 1$

Step 2.3.2

Add $3$ and $1$ .

$5 x = 4$

Step 2.4

Divide each term in $5 x = 4$ by $5$ and simplify.

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Step 2.4.1

Divide each term in $5 x = 4$ by $5$ .

$\frac{5 x}{5} = \frac{4}{5}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $5$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{4}{5}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{5}$

Decimal Form:

$x = 0.8$