Algebra Examples

$\log_{4} (7 x) + 2 \log_{4} (2) = 2$

Step 1

Simplify each term.

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Step 1.1

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

$\log_{4} (7 x) + 2 (\frac{1}{2}) = 2$

Step 1.2

Cancel the common factor of $2$ .

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Step 1.2.1

Cancel the common factor.

$\log_{4} (7 x) + 2 (\frac{1}{2}) = 2$

Step 1.2.2

Rewrite the expression.

$\log_{4} (7 x) + 1 = 2$

Step 2

Move all terms not containing $\log_{4} (7 x)$ to the right side of the equation.

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Step 2.1

Subtract $1$ from both sides of the equation.

$\log_{4} (7 x) = 2 - 1$

Step 2.2

Subtract $1$ from $2$ .

$\log_{4} (7 x) = 1$

Step 3

Rewrite $\log_{4} (7 x) = 1$ 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 $b^{y} = x$ .

$4^{1} = 7 x$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $7 x = 4^{1}$ .

$7 x = 4$

Step 4.2

Divide each term in $7 x = 4^{1}$ by $7$ and simplify.

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Step 4.2.1

Divide each term in $7 x = 4^{1}$ by $7$ .

$\frac{7 x}{7} = \frac{4^{1}}{7}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $7$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{4^{1}}{7}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4^{1}}{7}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Evaluate the exponent.

$x = \frac{4}{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{7}$

Decimal Form:

$x = 0.\overline{571428}$