Problem 61 - A Graphical Approach to College Algebra

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log x=\sqrt{\log x}$$

Step-by-Step Solution

Answer

The solutions are \( x = 1 \) and \( x = 10 \).

Step 1: Understand the Equation

The given equation is \( \log x = \sqrt{\log x} \). This implies that the logarithm of \( x \), \( \log x \), is equal to the square root of the logarithm of \( x \). We'll solve this by setting \( \log x = y \), simplifying the equation to \( y = \sqrt{y} \).

Step 2: Rewrite the Equation

Substituting \( y \) for \( \log x \), the equation becomes \( y = \sqrt{y} \). To eliminate the square root, square both sides: \( y^2 = y \).

Step 3: Rearrange into a Quadratic Form

The equation \( y^2 = y \) can be rewritten as \( y^2 - y = 0 \), a standard form for a quadratic equation.

Step 4: Factor the Quadratic Equation

Factor the quadratic equation \( y^2 - y = 0 \) as \( y(y - 1) = 0 \).

Step 5: Solve for y

Setting each factor to zero gives the solutions \( y = 0 \) and \( y = 1 \).

Step 6: Back-Substitute for x

Recall that \( y = \log x \). Thus the solutions are \( \log x = 0 \) and \( \log x = 1 \). Solving these for \( x \), we obtain \( x = 10^0 = 1 \) and \( x = 10^1 = 10 \).

Step 7: Verify Solutions Using a Calculator

Verify by checking: For \( x = 1 \), \( \log 1 = 0 \) and \( \sqrt{0} = 0 \). For \( x = 10 \), \( \log 10 = 1 \) and \( \sqrt{1} = 1 \). Both solutions satisfy the original equation.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. It is called 'quadratic' because the highest power of the variable \( x \) is 2. In our exercise, we transformed the equation \( y = \sqrt{y} \) into \( y^2 = y \), and then rearranged it to take the quadratic form \( y^2 - y = 0 \).Quadratic equations have two solutions, which can be found using various methods. Here, we used factorization, but other common methods include the quadratic formula and completing the square. Quadratic equations can represent various real-world scenarios, such as the trajectory of an object or the area of a rectangle, making it an essential concept to master.

Factorization

Factorization is the process of breaking down an expression into a product of simpler expressions or factors. In the case of the quadratic equation \( y^2 - y = 0 \), we can factor it by taking out the common factor \( y \), giving \( y(y - 1) = 0 \).Factorization is useful because it allows us to solve equations by setting each factor to zero. This gives us \( y = 0 \) and \( y - 1 = 0 \), leading to \( y = 0 \) and \( y = 1 \). By breaking problems into smaller parts, factorization can simplify complex algebraic expressions. It is a powerful tool used extensively in mathematics to simplify expressions and solve equations.

Exponential Functions

Exponential functions are mathematical functions of the form \( f(x) = a^x \), where the base \( a \) is a constant and the exponent \( x \) is a variable. They are fundamental in describing many natural phenomena, such as population growth, radioactive decay, and interest calculations.In our problem, when solving \( \log x = 0 \) and \( \log x = 1 \), the base of the logarithm is assumed to be 10. Thus, we convert each equation to exponential form: \( 10^0 = 1 \) and \( 10^1 = 10 \). Exponential functions have the property that they quickly increase or decrease depending on the base and the sign of the exponent. Here, they visually represent the solutions to the original logarithmic equation.

Verification of Solutions

Verification of solutions is an essential step when solving equations to ensure that all obtained solutions are correct. It involves substituting the solutions back into the original equation to check if they satisfy it.In our exercise, we derived the solutions \( x = 1 \) and \( x = 10 \) for the equation \( \log x = \sqrt{\log x} \). To verify:

• For \( x = 1 \), \( \log 1 = 0 \) and \( \sqrt{0} = 0 \), confirming that it satisfies the equation.
• For \( x = 10 \), \( \log 10 = 1 \) and \( \sqrt{1} = 1 \), also confirming it satisfies the equation.

Verification helps catch errors and validate that our mathematical manipulations didn't introduce incorrect solutions. It is a crucial step in any math problem to confirm the validity of answers.