Trigonometry Examples

$\log_{5} (y) - 4 (\log_{5} (r) + 2 \log_{5} (t))$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{5} (t)$ by moving $2$ inside the logarithm.

$\log_{5} (y) - 4 (\log_{5} (r) + \log_{5} (t^{2}))$

Step 1.2

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

$\log_{5} (y) - 4 \log_{5} (r t^{2})$

Step 1.3

Simplify $- 4 \log_{5} (r t^{2})$ by moving $4$ inside the logarithm.

$\log_{5} (y) - \log_{5} ({(r t^{2})}^{4})$

Step 1.4

Apply the product rule to $r t^{2}$ .

$\log_{5} (y) - \log_{5} (r^{4} {(t^{2})}^{4})$

Step 1.5

Multiply the exponents in ${(t^{2})}^{4}$ .

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Step 1.5.1

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

$\log_{5} (y) - \log_{5} (r^{4} t^{2 \cdot 4})$

Step 1.5.2

Multiply $2$ by $4$ .

$\log_{5} (y) - \log_{5} (r^{4} t^{8})$

Step 2

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

$\log_{5} (\frac{y}{r^{4} t^{8}})$