Solve the Logarithmic Equation: Log_4(3x^2+8x-10) = Log_4(-x^2-x+12.5)

Video Solution

Solution Steps

00:00 Solve

00:08 We want to find the domain

00:19 We'll use the roots formula to find the possible solutions

00:31 Let's calculate and solve

01:03 Remember that there are 2 possible solutions, positive and negative

01:34 These are the solutions, let's find the positive domain

01:46 This is the domain according to the first log

01:59 Now let's use the same method for the second log

02:04 We'll use the roots formula to find the possible solutions

02:14 Let's calculate and solve

02:34 Remember that there are 2 possible solutions, positive and negative

02:54 These are the solutions, let's find the positive domain

03:09 Now let's find the complete domain

03:24 This is the domain

03:44 We'll use the formula for log division, we'll get the quotient log

03:54 Let's use this formula in our exercise

04:19 Let's solve according to the log definition

04:24 Any number to the power of 0 always equals 1

04:29 Let's multiply by the denominator to eliminate the fraction

04:57 Let's arrange the equation

05:26 We'll use the roots formula to find the possible solutions

05:40 Let's calculate and solve

06:24 Let's note the 2 possible solutions

06:50 According to the domain, we'll find the solution

06:55 And this is the solution to the question

Step-by-Step Solution

To solve this problem, we'll apply the following steps:

Step 1: Use logarithmic properties to rewrite $\log_4(3x^2+8x-10) - \log_4(-x^2-x+12.5) = 0$ as a single logarithm: $\log_4 \left( \frac{3x^2 + 8x - 10}{-x^2 - x + 12.5} \right) = 0$ .
Step 2: Recognize that if $\log_4 \left( \frac{3x^2 + 8x - 10}{-x^2 - x + 12.5} \right) = 0$ , then $\frac{3x^2 + 8x - 10}{-x^2 - x + 12.5} = 4^0 = 1$ .
Step 3: Set up the equation: $3x^2 + 8x - 10 = -x^2 - x + 12.5$ .
Step 4: Rearrange the equation to: $3x^2 + 8x - 10 + x^2 + x - 12.5 = 0$ .
Step 5: Simplify to: $4x^2 + 9x - 22.5 = 0$ .
Step 6: Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 4$ , $b = 9$ , $c = -22.5$ .
Step 7: Calculate: $b^2 - 4ac = 9^2 - 4 \times 4 \times (-22.5) = 81 + 360 = 441$ .
Step 8: Find $x$ : $x = \frac{-9 \pm \sqrt{441}}{8} = \frac{-9 \pm 21}{8}$ .
Step 9: Compute the roots: $x_1 = \frac{-9 + 21}{8} = \frac{12}{8} = 1.5$ and $x_2 = \frac{-9 - 21}{8} = \frac{-30}{8} = -3.75$ .
Step 10: Verify these solutions satisfy the domain of original logarithmic expressions by substituting back into $3x^2 + 8x - 10 > 0$ and $-x^2 - x + 12.5 > 0$ .

Therefore, the solutions to the problem are $x = -3.75, 1.5$ .

The correct choice from the provided options is: $-3.75, 1.5$ .