Algebra Examples

$\frac{1}{2} \cdot \log (x) + \log (4) = 2$

Step 1

Simplify the left side.

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Step 1.1

Combine $\frac{1}{2}$ and $\log (x)$ .

$\frac{\log (x)}{2} + \log (4) = 2$

Step 2

Simplify the left side.

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Step 2.1

Simplify $\frac{\log (x)}{2} + \log (4)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Rewrite $\frac{\log (x)}{2}$ as $\frac{1}{2} \log (x)$ .

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

Step 2.1.1.2

Simplify $\frac{1}{2} \log (x)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log (x^{\frac{1}{2}}) + \log (4) = 2$

Step 2.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (x^{\frac{1}{2}} \cdot 4) = 2$

Step 2.1.3

Move $4$ to the left of $x^{\frac{1}{2}}$ .

$\log (4 x^{\frac{1}{2}}) = 2$

Step 3

Rewrite $\log (4 x^{\frac{1}{2}}) = 2$ 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 $b^{y} = x$ .

$10^{2} = 4 x^{\frac{1}{2}}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $4 x^{\frac{1}{2}} = 10^{2}$ .

$4 x^{\frac{1}{2}} = 10^{2}$

Step 4.2

Divide each term in $4 x^{\frac{1}{2}} = 10^{2}$ by $4$ and simplify.

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Step 4.2.1

Divide each term in $4 x^{\frac{1}{2}} = 10^{2}$ by $4$ .

$\frac{4 x^{\frac{1}{2}}}{4} = \frac{10^{2}}{4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor.

$\frac{4 x^{\frac{1}{2}}}{4} = \frac{10^{2}}{4}$

Step 4.2.2.2

Divide $x^{\frac{1}{2}}$ by $1$ .

$x^{\frac{1}{2}} = \frac{10^{2}}{4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raise $10$ to the power of $2$ .

$x^{\frac{1}{2}} = \frac{100}{4}$

Step 4.2.3.2

Divide $100$ by $4$ .

$x^{\frac{1}{2}} = 25$

Step 4.3

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{2}})}^{2} = 25^{2}$

Step 4.4

Simplify the exponent.

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 4.4.1.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 4.4.1.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 25^{2}$

Step 4.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 4.4.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 25^{2}$

Step 4.4.1.1.1.2.2

Rewrite the expression.

$x^{1} = 25^{2}$

Step 4.4.1.1.2

Simplify.

$x = 25^{2}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Raise $25$ to the power of $2$ .

$x = 625$