Algebra Examples

$\log_{7} (y) - 2 (\log_{7} (x) + \frac{1}{2} \log_{7} (z))$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \log_{7} (z)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{7} (y) - 2 (\log_{7} (x) + \log_{7} (z^{\frac{1}{2}}))$

Step 1.2

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

$\log_{7} (y) - 2 \log_{7} (x z^{\frac{1}{2}})$

Step 1.3

Simplify $- 2 \log_{7} (x z^{\frac{1}{2}})$ by moving $2$ inside the logarithm.

$\log_{7} (y) - \log_{7} ({(x z^{\frac{1}{2}})}^{2})$

Step 1.4

Apply the product rule to $x z^{\frac{1}{2}}$ .

$\log_{7} (y) - \log_{7} (x^{2} {(z^{\frac{1}{2}})}^{2})$

Step 1.5

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

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Step 1.5.1

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

$\log_{7} (y) - \log_{7} (x^{2} z^{\frac{1}{2} \cdot 2})$

Step 1.5.2

Cancel the common factor of $2$ .

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Step 1.5.2.1

Cancel the common factor.

$\log_{7} (y) - \log_{7} (x^{2} z^{\frac{1}{2} \cdot 2})$

Step 1.5.2.2

Rewrite the expression.

$\log_{7} (y) - \log_{7} (x^{2} z^{1})$

Step 1.6

Simplify.

$\log_{7} (y) - \log_{7} (x^{2} z)$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{7} (\frac{y}{x^{2} z})$