Problem 11 - Algebra 2

Question

Solve each equation. Check your solutions. \(\log _{10} a+\log _{10}(a+21)=2\)

Step-by-Step Solution

Answer

The solution is \( a = 4 \).

Step 1: Apply the Product Property of Logarithms

We start by using the product property of logarithms, which states that \( \log_b(x) + \log_b(y) = \log_b(xy) \). In our equation, this gives us \( \log_{10}(a(a + 21)) = 2 \).

Step 2: Convert the Logarithmic Equation to an Exponential Equation

Next, we convert the equation from logarithmic form to exponential form. Given \( \log_{10}(b) = c \), this translates to \( b = 10^c \). Applying this to our equation, we get: \( a(a + 21) = 10^2 \), which simplifies to \( a(a + 21) = 100 \).

Step 3: Solve the Quadratic Equation

Expand the left side and rewrite the equation as a standard quadratic equation: \( a^2 + 21a - 100 = 0 \). To solve this, we will use the quadratic formula, \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), with \( a = 1, b = 21, c = -100 \).

Step 4: Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \). For our equation, this is \( 21^2 - 4 \times 1 \times (-100) = 441 + 400 = 841 \). Since the discriminant is a perfect square, the roots will be real and rational.

Step 5: Use the Quadratic Formula

Substitute the values into the quadratic formula: \( a = \frac{-21 \pm \sqrt{841}}{2} \). This gives us \( a = \frac{-21 \pm 29}{2} \).

Step 6: Find the Possible Values for a

Calculate the two possible solutions: \( a = \frac{8}{2} = 4 \) and \( a = \frac{-50}{2} = -25 \). However, since \( a \) must be positive (because we are dealing with logarithms), we discard \( a = -25 \). Thus, \( a = 4 \) is the only valid solution.

Step 7: Verify the Solution

We verify the solution by substituting \( a = 4 \) back into the original equation: \( \log_{10}(4) + \log_{10}(25) = \log_{10}(100) = 2 \). The solution satisfies the equation, confirming \( a = 4 \) is correct.

Key Concepts

Product Property of Logarithms

The product property of logarithms is a powerful tool that makes simplifying some logarithmic expressions much easier. This property states that the logarithm of a product is the sum of the logarithms of its factors. In mathematical terms, for any positive numbers \( x \) and \( y \), and any base \( b \), the property is written as:

\( \log_b(xy) = \log_b(x) + \log_b(y) \)

In our equation, \( \log_{10}(a) + \log_{10}(a + 21) = 2 \), we can use this property to combine the two logarithmic terms:

\( \log_{10}(a(a + 21)) = 2 \)

This combination allows us to deal with a single logarithmic expression instead of multiple ones, simplifying the solving process significantly.
After converting and simplifying using this property, we can move on to solve the equation more easily.

Quadratic Equation

A quadratic equation is a second-order polynomial equation in a single variable \( x \), with the general form:

\( ax^2 + bx + c = 0 \)

The equation from our problem transformed into this form is:

\( a^2 + 21a - 100 = 0 \)

Recognizing this as a quadratic equation is crucial because it allows us to use specific methods to find its roots, or solutions. One of the most effective ways is by using the quadratic formula, which provides a standardized approach to solving quadratic equations.
The quadratic formula ensures that we can find solutions even when factoring is complicated or inefficient.

Discriminant

The discriminant is a vital component of the quadratic formula and helps us understand the nature of the solutions of the quadratic equation. Given a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( D \) is calculated as:

\( D = b^2 - 4ac \)

For our specific equation, it looks like this:

\( 21^2 - 4(1)(-100) = 841 \)

The discriminant tells us:

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), there is exactly one real root (a repeated or double root).
• If \( D < 0 \), the roots are complex (not real).

Since our discriminant is 841, a positive perfect square, we know this quadratic equation has two distinct, real, and rational roots. This insight is essential before proceeding to use the quadratic formula.

Logarithmic Form

The logarithmic form of an equation is a way to express exponential relationships. A logarithm answers the question: 'To what power must the base be raised, to produce this number?' For instance, in our equation,

\( \log_{10}(a^2 + 21a) = 2 \)

is the logarithmic form.
Understanding this form is crucial because it allows us to approach problems needing a conversion between an exponential and a logarithmic perspective. In solving our problem, switching from the logarithmic form to exponential form helps us translate the relationship into a more tractable multiplication problem.

Exponential Form

An exponential form is a mathematical expression where numbers are expressed with a base raised to a power. Converting an equation from logarithmic to exponential form involves rewriting it such that the base with its power gives a result. For the given problem's logarithmic equation

\( \log_{10}(a(a + 21)) = 2 \)

we convert it to its exponential form as:

\( a(a + 21) = 10^2 \)

or simply:

\( a(a + 21) = 100 \)

In exponential form, the equation showcases a clearer path to further simplify and solve the problem. Recognizing and performing this conversion allows us to attack the equation with a variety of algebraic techniques, including factorization and the use of quadratic solutions. Exponential form serves as a bridge from logarithmic equations to usable mathematical relationships amenable to further manipulation.