# 01 - Solution of Logarithmic Equations

Solve for x from the following:

- $\log_6 (x - 2) + \log_6 (x + 3) = 1$

- $x^{\log x} = 10\,000$

Solve for x from the following:

- $\log_6 (x - 2) + \log_6 (x + 3) = 1$

- $x^{\log x} = 10\,000$

**Problem**

Solve for $x$ and $y$ from the given system of equations.

$\dfrac{3}{x^2} - \dfrac{4}{y^2} = 2$ ← Equation (1)

$\dfrac{5}{x^2} - \dfrac{3}{y^2} = \dfrac{17}{4}$ ← Equation (2)

**Problem**

Solve for $x$ and $y$ from the given system of equations.

$x^2y + y = 17$ ← Equation (1)

$x^4y^2 + y^2 = 257$ ← Equation (2)

**Problem**

Solve for x and y from the given system of equations.

$x + 2y = 6$ ← Equation (1)

$\sqrt{x} + \sqrt{y} = 3$ ← Equation (2)

Solve for $x$ from the following equations:

- $\left( \dfrac{x^2 - 15}{x} \right)^2 - 16\left( \dfrac{15 - x^2}{x} \right) + 28 = 0$

- $\dfrac{x}{\sqrt{x} + \sqrt{9 - x}} + \dfrac{x}{\sqrt{x} - \sqrt{9 - x}} = \dfrac{24}{\sqrt{x}}$

Determine the value of $x$ from the following equations:

- $\sqrt{(4 - x^2)^3} + 3x^2\sqrt{4 - x^2} = 0$

- $\dfrac{1}{3x - 2} - \dfrac{8}{\sqrt{3x - 2}} = 9$

Solve for $x$ from the following equations

- $\sqrt{3 - x} + \sqrt{4 - 2x} = \sqrt{3 - 3x}$

- $\sqrt{\dfrac{2x + 4}{x - 5}} + 8\sqrt{\dfrac{x - 5}{2x + 4}} = 6$

**Problem 3**

A man bought a cellphone and laptop for P50,000.00, paying three times as much for the laptop as for the cellphone. How much did each cost?

**Problem 4**

Two boys, counting their money, found that together they had P372, and that Rody had ﬁve times as much as Mar. How much had each?

**Problem 1**

The sum of two numbers is 120. If the greater number is four times the less, what are the numbers?

**Problem 2**

The greater of two numbers is twice the less, and the sum of the numbers is 96. What are the numbers?

**Problem**

What is the equation of the curve passing through the point (3, -2) and having a slope at any point (x, y) equal to (x^{2} + y^{2}) / (y^{3} - 2xy)?

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