Precalculus Examples

$\log (x - 3) + \log (2 x - 5) = \log_{3} (1) + 6^{\log_{6} (2)}$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

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

$\log ((x - 3) (2 x - 5)) = \log_{3} (1) + 6^{\log_{6} (2)}$

Step 1.2

Expand $(x - 3) (2 x - 5)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$\log (x (2 x - 5) - 3 (2 x - 5)) = \log_{3} (1) + 6^{\log_{6} (2)}$

Step 1.2.2

Apply the distributive property.

$\log (x (2 x) + x \cdot - 5 - 3 (2 x - 5)) = \log_{3} (1) + 6^{\log_{6} (2)}$

Step 1.2.3

Apply the distributive property.

$\log (x (2 x) + x \cdot - 5 - 3 (2 x) - 3 \cdot - 5) = \log_{3} (1) + 6^{\log_{6} (2)}$

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$\log (2 x \cdot x + x \cdot - 5 - 3 (2 x) - 3 \cdot - 5) = \log_{3} (1) + 6^{\log_{6} (2)}$

Step 1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.3.1.2.1

Move $x$ .

$\log (2 (x \cdot x) + x \cdot - 5 - 3 (2 x) - 3 \cdot - 5) = \log_{3} (1) + 6^{\log_{6} (2)}$

Step 1.3.1.2.2

Multiply $x$ by $x$ .

$\log (2 x^{2} + x \cdot - 5 - 3 (2 x) - 3 \cdot - 5) = \log_{3} (1) + 6^{\log_{6} (2)}$

Step 1.3.1.3

Move $- 5$ to the left of $x$ .

$\log (2 x^{2} - 5 \cdot x - 3 (2 x) - 3 \cdot - 5) = \log_{3} (1) + 6^{\log_{6} (2)}$

Step 1.3.1.4

Multiply $2$ by $- 3$ .

$\log (2 x^{2} - 5 x - 6 x - 3 \cdot - 5) = \log_{3} (1) + 6^{\log_{6} (2)}$

Step 1.3.1.5

Multiply $- 3$ by $- 5$ .

$\log (2 x^{2} - 5 x - 6 x + 15) = \log_{3} (1) + 6^{\log_{6} (2)}$

Step 1.3.2

Subtract $6 x$ from $- 5 x$ .

$\log (2 x^{2} - 11 x + 15) = \log_{3} (1) + 6^{\log_{6} (2)}$

Step 2

Simplify the right side.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Logarithm base $3$ of $1$ is $0$ .

$\log (2 x^{2} - 11 x + 15) = 0 + 6^{\log_{6} (2)}$

Step 2.1.2

Exponentiation and log are inverse functions.

$\log (2 x^{2} - 11 x + 15) = 0 + 2$

Step 2.2

Add $0$ and $2$ .

$\log (2 x^{2} - 11 x + 15) = 2$

Step 3

Rewrite $\log (2 x^{2} - 11 x + 15) = 2$ 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 $b^{y} = x$ .

$10^{2} = 2 x^{2} - 11 x + 15$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $2 x^{2} - 11 x + 15 = 10^{2}$ .

$2 x^{2} - 11 x + 15 = 10^{2}$

Step 4.2

Raise $10$ to the power of $2$ .

$2 x^{2} - 11 x + 15 = 100$

Step 4.3

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

Tap for more steps...

Step 4.3.1

Subtract $100$ from both sides of the equation.

$2 x^{2} - 11 x + 15 - 100 = 0$

Step 4.3.2

Subtract $100$ from $15$ .

$2 x^{2} - 11 x - 85 = 0$

Step 4.4

Use the quadratic formula to find the solutions.

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

Step 4.5

Substitute the values $a = 2$ , $b = - 11$ , and $c = - 85$ into the quadratic formula and solve for $x$ .

$\frac{11 \pm \sqrt{{(- 11)}^{2} - 4 \cdot (2 \cdot - 85)}}{2 \cdot 2}$

Step 4.6

Simplify.

Tap for more steps...

Step 4.6.1

Simplify the numerator.

Tap for more steps...

Step 4.6.1.1

Raise $- 11$ to the power of $2$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 2 \cdot - 85}}{2 \cdot 2}$

Step 4.6.1.2

Multiply $- 4 \cdot 2 \cdot - 85$ .

Tap for more steps...

Step 4.6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{11 \pm \sqrt{121 - 8 \cdot - 85}}{2 \cdot 2}$

Step 4.6.1.2.2

Multiply $- 8$ by $- 85$ .

$x = \frac{11 \pm \sqrt{121 + 680}}{2 \cdot 2}$

Step 4.6.1.3

Add $121$ and $680$ .

$x = \frac{11 \pm \sqrt{801}}{2 \cdot 2}$

Step 4.6.1.4

Rewrite $801$ as $3^{2} \cdot 89$ .

Tap for more steps...

Step 4.6.1.4.1

Factor $9$ out of $801$ .

$x = \frac{11 \pm \sqrt{9 (89)}}{2 \cdot 2}$

Step 4.6.1.4.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{11 \pm \sqrt{3^{2} \cdot 89}}{2 \cdot 2}$

Step 4.6.1.5

Pull terms out from under the radical.

$x = \frac{11 \pm 3 \sqrt{89}}{2 \cdot 2}$

Step 4.6.2

Multiply $2$ by $2$ .

$x = \frac{11 \pm 3 \sqrt{89}}{4}$

Step 4.7

The final answer is the combination of both solutions.

$x = \frac{11 + 3 \sqrt{89}}{4}, \frac{11 - 3 \sqrt{89}}{4}$

Step 5

Exclude the solutions that do not make $\log (x - 3) + \log (2 x - 5) = \log_{3} (1) + 6^{\log_{6} (2)}$ true.

$x = \frac{11 + 3 \sqrt{89}}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11 + 3 \sqrt{89}}{4}$

Decimal Form:

$x = 9.82548584 \dots$