Precalculus Examples

$\log_{2} (t + 1) - \log_{2} (t - 1) = 3$

Step 1

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

$\log_{2} (\frac{t + 1}{t - 1}) = 3$

Step 2

Rewrite $\log_{2} (\frac{t + 1}{t - 1}) = 3$ 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 $b^{y} = x$ .

$2^{3} = \frac{t + 1}{t - 1}$

Step 3

Cross multiply to remove the fraction.

$t + 1 = 2^{3} (t - 1)$

Step 4

Simplify $2^{3} (t - 1)$ .

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Step 4.1

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

$t + 1 = 8 (t - 1)$

Step 4.2

Apply the distributive property.

$t + 1 = 8 t + 8 \cdot - 1$

Step 4.3

Multiply $8$ by $- 1$ .

$t + 1 = 8 t - 8$

Step 5

Move all terms containing $t$ to the left side of the equation.

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Step 5.1

Subtract $8 t$ from both sides of the equation.

$t + 1 - 8 t = - 8$

Step 5.2

Subtract $8 t$ from $t$ .

$- 7 t + 1 = - 8$

Step 6

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

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Step 6.1

Subtract $1$ from both sides of the equation.

$- 7 t = - 8 - 1$

Step 6.2

Subtract $1$ from $- 8$ .

$- 7 t = - 9$

Step 7

Divide each term in $- 7 t = - 9$ by $- 7$ and simplify.

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Step 7.1

Divide each term in $- 7 t = - 9$ by $- 7$ .

$\frac{- 7 t}{- 7} = \frac{- 9}{- 7}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 7$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 7 t}{- 7} = \frac{- 9}{- 7}$

Step 7.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 9}{- 7}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Dividing two negative values results in a positive value.

$t = \frac{9}{7}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$t = \frac{9}{7}$

Decimal Form:

$t = 1.\overline{285714}$

Mixed Number Form:

$t = 1 \frac{2}{7}$