Calculus Examples

$\log_{8} (z - 6) = 2 - \log_{8} (z + 15)$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$\log_{8} (z - 6) + \log_{8} (z + 15) = 2$

Step 2

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

$\log_{8} ((z - 6) (z + 15)) = 2$

Step 3

Expand $(z - 6) (z + 15)$ using the FOIL Method.

Tap for more steps...

Step 3.1

Apply the distributive property.

$\log_{8} (z (z + 15) - 6 (z + 15)) = 2$

Step 3.2

Apply the distributive property.

$\log_{8} (z \cdot z + z \cdot 15 - 6 (z + 15)) = 2$

Step 3.3

Apply the distributive property.

$\log_{8} (z \cdot z + z \cdot 15 - 6 z - 6 \cdot 15) = 2$

Step 4

Simplify and combine like terms.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Multiply $z$ by $z$ .

$\log_{8} (z^{2} + z \cdot 15 - 6 z - 6 \cdot 15) = 2$

Step 4.1.2

Move $15$ to the left of $z$ .

$\log_{8} (z^{2} + 15 \cdot z - 6 z - 6 \cdot 15) = 2$

Step 4.1.3

Multiply $- 6$ by $15$ .

$\log_{8} (z^{2} + 15 z - 6 z - 90) = 2$

Step 4.2

Subtract $6 z$ from $15 z$ .

$\log_{8} (z^{2} + 9 z - 90) = 2$

Step 5

Rewrite $\log_{8} (z^{2} + 9 z - 90) = 2$ 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 $b^{y} = x$ .

$8^{2} = z^{2} + 9 z - 90$

Step 6

Solve for $z$ .

Tap for more steps...

Step 6.1

Rewrite the equation as $z^{2} + 9 z - 90 = 8^{2}$ .

$z^{2} + 9 z - 90 = 8^{2}$

Step 6.2

Raise $8$ to the power of $2$ .

$z^{2} + 9 z - 90 = 64$

Step 6.3

Move all terms to the left side of the equation and simplify.

Tap for more steps...

Step 6.3.1

Subtract $64$ from both sides of the equation.

$z^{2} + 9 z - 90 - 64 = 0$

Step 6.3.2

Subtract $64$ from $- 90$ .

$z^{2} + 9 z - 154 = 0$

Step 6.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.5

Substitute the values $a = 1$ , $b = 9$ , and $c = - 154$ into the quadratic formula and solve for $z$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (1 \cdot - 154)}}{2 \cdot 1}$

Step 6.6

Simplify.

Tap for more steps...

Step 6.6.1

Simplify the numerator.

Tap for more steps...

Step 6.6.1.1

Raise $9$ to the power of $2$ .

$z = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1 \cdot - 154}}{2 \cdot 1}$

Step 6.6.1.2

Multiply $- 4 \cdot 1 \cdot - 154$ .

Tap for more steps...

Step 6.6.1.2.1

Multiply $- 4$ by $1$ .

$z = \frac{- 9 \pm \sqrt{81 - 4 \cdot - 154}}{2 \cdot 1}$

Step 6.6.1.2.2

Multiply $- 4$ by $- 154$ .

$z = \frac{- 9 \pm \sqrt{81 + 616}}{2 \cdot 1}$

Step 6.6.1.3

Add $81$ and $616$ .

$z = \frac{- 9 \pm \sqrt{697}}{2 \cdot 1}$

Step 6.6.2

Multiply $2$ by $1$ .

$z = \frac{- 9 \pm \sqrt{697}}{2}$

Step 6.7

The final answer is the combination of both solutions.

$z = - \frac{9 - \sqrt{697}}{2}, - \frac{9 + \sqrt{697}}{2}$

Step 7

Exclude the solutions that do not make $\log_{8} (z - 6) = 2 - \log_{8} (z + 15)$ true.

$z = - \frac{9 - \sqrt{697}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$z = - \frac{9 - \sqrt{697}}{2}$

Decimal Form:

$z = 8.70037878 \dots$