Solving Logarithmic Equation

Category: Algebra

"Published in Newark, California, USA"

Find the solution set of the equation:

Solution:

The first thing that we have to do is to transpose the constant at the right side of the equation as follows

Since the left side of the equation have the same base which is x, then we can combine and apply the laws of logarithm as follows

Take the inverse logarithm on both sides of the equation to the base x as follows

The roots of the equation are: x = 3 and x = -1.

Next, check the roots of the equation by substituting the value of x to the given equation as follows

If x = 3, then

                        x = 3 is the root of the equation

If x = -1, then

                       x = -1 is an extraneous root

Therefore, the answer is x = 3. 

Evaluating Inverse Trigonometric Functions

Category: Trigonometry

"Published in Newark, California, USA"

Evaluate, using principal values for the inverse functions involved

Solution:

The given equation above involves with inverse trigonometric functions. Each inverse trigonometric functions represents an angle. We can get the value of the above equation without using a calculator as follows

Let 

Express the above trigonometric function in a right triangle

Photo by Math Principles in Everyday Life

From the figure above, we can find the other trigonometric functions as follows

Let 

Express the above trigonometric function in a right triangle

Picture by Math Principles in Everyday Life

From the figure above, we can find the other trigonometric functions as follows

Now, consider the given equation

Rewrite the above equation as follows

Finally, substitute the value of each trigonometric functions to the above equation as follows

Algebraic Operations - Exponential Fractions

Category: Algebra

"Published in Newark, California, USA"

Perform the indicated operations and simplify:

Solution:

Consider the above equation

Since the terms in the numerator and denominator are monomial and have the same term which are 3, x, and y, then we can apply the laws of exponents as follows