Algebra Examples

$2 \log_{4} (5 x) = 3$

Step 1

Divide each term in $2 \log_{4} (5 x) = 3$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 \log_{4} (5 x) = 3$ by $2$ .

$\frac{2 \log_{4} (5 x)}{2} = \frac{3}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 \log_{4} (5 x)}{2} = \frac{3}{2}$

Step 1.2.1.2

Divide $\log_{4} (5 x)$ by $1$ .

$\log_{4} (5 x) = \frac{3}{2}$

Step 2

Rewrite $\log_{4} (5 x) = \frac{3}{2}$ 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 $b^{y} = x$ .

$4^{\frac{3}{2}} = 5 x$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $5 x = 4^{\frac{3}{2}}$ .

$5 x = 4^{\frac{3}{2}}$

Step 3.2

Divide each term in $5 x = 4^{\frac{3}{2}}$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 x = 4^{\frac{3}{2}}$ by $5$ .

$\frac{5 x}{5} = \frac{4^{\frac{3}{2}}}{5}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{4^{\frac{3}{2}}}{5}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4^{\frac{3}{2}}}{5}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify the numerator.

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Step 3.2.3.1.1

Rewrite $4$ as $2^{2}$ .

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

Step 3.2.3.1.2

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

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

Step 3.2.3.1.3

Cancel the common factor of $2$ .

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Step 3.2.3.1.3.1

Cancel the common factor.

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

Step 3.2.3.1.3.2

Rewrite the expression.

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

Step 3.2.3.1.4

Raise $2$ to the power of $3$ .

$x = \frac{8}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{8}{5}$

Decimal Form:

$x = 1.6$

Mixed Number Form:

$x = 1 \frac{3}{5}$