The Sum of Logarithms: Resulting in a quadratic equation

Defined domain

x>0

x+1>0

x>-1

$\log7x+\log\left(x+1\right)-\log7=\log2x-\log x$

$\log\frac{7x\cdot\left(x+1\right)}{7}=\log\frac{2x}{x}$

We reduce by: $7$ and by $X$

$x\left(x+1\right)=2$

$x^2+x-2=0$

$\left(x+2\right)\left(x-1\right)=0$

$x+2=0$

$x=-2$

Undefined domain x>0

$x-1=0$

$x=1$

Defined domain

To solve the equation: $\log_8 9 - \log_8 3 + \log_4 x^2 = \log_8 1.5 + \log_8 2 + \log_4 (-x^2 - 11x - 9)$, we proceed as follows:

Step 1: Simplify Both Sides
On the left-hand side (LHS), apply logarithmic subtraction: $\log_8 \left(\frac{9}{3}\right) + \log_4 x^2 = \log_8 3 + \log_4 x^2$.
Note $\log_8 3$ remains and convert $\log_4 x^2$ using the base switch to $8$:
$\log_4 x^2 = 2\log_4 x = 2 \times \frac{\log_8 x}{\log_8 2^2} = \frac{\log_8 x}{\log_8 2}$.
Thus, the LHS combines into:
$\log_8 3 + \frac{2\log_8 x}{\log_8 4}$ (because $\log_4 x^2 = 2 \log_4 x$).

On the right-hand side (RHS):
Combine: $\log_8 (1.5 \times 2) = \log_8 3$.
Also apply for $\log_4$ term:
$\log_4 (-x^2 - 11x - 9) = \frac{\log_8 (-x^2 - 11x - 9)}{\log_8 4}$.

Step 2: Equalize Both Sides
Equate LHS and RHS logarithmic expressions:
$\log_8 3 + \frac{2\log_8 x}{\log_8 4} = \log_8 3 + \frac{\log_8 (-x^2 - 11x - 9)}{\log_8 4}$.
The $\log_8 3$ cancels out on both sides, leaving:
$\frac{2\log_8 x}{\log_8 4} = \frac{\log_8 (-x^2 - 11x - 9)}{\log_8 4}$.

Step 3: Solve for $x$
Since the denominators are equal, set the numerators equal:
$2\log_8 x = \log_8 (-x^2 - 11x - 9)$.
Translate this into an exponential equation:
$(x^2)^2 = -x^2 - 11x - 9$ or
$8^{2\log_8 x} = -x^2 - 11x - 9$.
Let $y = x$, solve the resulting quadratic equation:
$x^2 = -x^2 - 11x - 9$.
Then, finding valid $x$ by allowing roots of polynomial calculations should yield laws consistency:
$-x^2 - 11x - 9 = 0$ or rather substituting potential values. After appropriate checks:

The valid $x$ that satisfies the problem is thus $x = -4.5$.

To solve the given equation, follow these steps:

We start with the expression:

$\frac{2\ln4}{\ln5} + \frac{1}{\log_{(x^2+8)}5} = \log_5(7x^2+9x)$

Use the change-of-base formula to rewrite everything in terms of natural logarithms:

$\frac{2\ln4}{\ln5} + \frac{\ln(x^2+8)}{\ln5} = \frac{\ln(7x^2+9x)}{\ln5}$

Multiplying the entire equation by $\ln 5$ to eliminate the denominators:

$2\ln4 + \ln(x^2+8) = \ln(7x^2+9x)$

By properties of logarithms (namely the product and power laws), combine the left side using the addition property:

$\ln(4^2(x^2+8)) = \ln(7x^2+9x)$

$\ln(16x^2 + 128) = \ln(7x^2 + 9x)$

Since the natural logarithm function is one-to-one, equate the arguments:

$16x^2 + 128 = 7x^2 + 9x$

Rearrange this into a standard form of a quadratic equation:

$9x^2 - 9x + 128 = 0$

Attempt to solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where $a = 9$, $b = -9$, and $c = -128$.

Calculate the discriminant:

$b^2 - 4ac = (-9)^2 - 4(9)(-128) = 81 + 4608$

$= 4689$

The discriminant is positive, suggesting real solutions should exist, however, verification against the domain constraints of logarithms (arguments must be positive) is needed.

After solving $9x^2 - 9x + 128 = 0$, the following is noted:

The polynomial does not yield any $x$ values in domains valid for the original logarithmic arguments.

Cross-verify the potential solutions against original conditions:

• For $\ln(x^2+8)$: Requires $x^2 + 8 > 0$, valid as $x$ values are always real.
• For $\ln(7x^2+9x)$: Requires $7x^2+9x > 0$, indicating constraints on $x$.

Solutions obtained do not satisfy these together within the purview of the rational roots and ultimately render no real value for $x$.

Therefore, the solution to the problem is: There is no solution.