Trigonometry Examples

$\log_{b} (t^{2} - 3 t) - \log_{b} (t) = \log_{b} (3)$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{b} (\frac{t^{2} - 3 t}{t}) = \log_{b} (3)$

Step 1.2

Factor $t$ out of $t^{2} - 3 t$ .

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Step 1.2.1

Factor $t$ out of $t^{2}$ .

$\log_{b} (\frac{t \cdot t - 3 t}{t}) = \log_{b} (3)$

Step 1.2.2

Factor $t$ out of $- 3 t$ .

$\log_{b} (\frac{t \cdot t + t \cdot - 3}{t}) = \log_{b} (3)$

Step 1.2.3

Factor $t$ out of $t \cdot t + t \cdot - 3$ .

$\log_{b} (\frac{t (t - 3)}{t}) = \log_{b} (3)$

Step 1.3

Cancel the common factor of $t$ .

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Step 1.3.1

Cancel the common factor.

$\log_{b} (\frac{t (t - 3)}{t}) = \log_{b} (3)$

Step 1.3.2

Divide $t - 3$ by $1$ .

$\log_{b} (t - 3) = \log_{b} (3)$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$t - 3 = 3$

Step 3

Move all terms not containing $t$ to the right side of the equation.

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Step 3.1

Add $3$ to both sides of the equation.

$t = 3 + 3$

Step 3.2

Add $3$ and $3$ .

$t = 6$