Problem 14 - Precalculus

Question

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$10^{\log \left(3 x^{2}+2 x+1\right)}$$

Step-by-Step Solution

Answer

The expression simplifies to \( 3x^2 + 2x + 1 \).

Step 1: Understanding the Expression

We start with the expression given in the problem: \[ 10^{\log(3x^2 + 2x + 1)} \] This expression involves an exponentiation of a function inside a logarithm.

Step 2: Apply the Property of Logarithms and Exponents

Recall the property that states \( a^{\log_a(b)} = b \). This property applies when the base of the logarithm matches the base of the exponentiation. Here, the base is 10, so the expression simplifies to:\[ 10^{\log(3x^2 + 2x + 1)} = 3x^2 + 2x + 1 \]This is because the properties of logarithms allow the logarithm and the exponentiation to "cancel out" when the base of the exponent matches the base of the logarithm.

Step 3: Conclusion

From the simplification in the previous step, we conclude that:\[ 10^{\log(3x^2 + 2x + 1)} = 3x^2 + 2x + 1 \]The expression reduces directly to the polynomial inside the logarithm.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers: a base and an exponent. It tells us how many times the base is to be multiplied by itself. For example, in the expression \(2^3\), 2 is the base and 3 is the exponent, and it equals \(2 \times 2 \times 2 = 8\).
Exponentiation is a fundamental operation in algebra and appears often in problems involving growth, area calculations, and in many scientific contexts.

• When you see an expression like \(b^n\), it means you're multiplying **b** by itself **n** times.
• The exponent can be any number: positive, negative, or even a fraction. However, the concept remains the same. It dictates repeated multiplication.

The challenges within exponentiation often lie in handling large numbers or simplifying expressions involving variable exponents.
You solve problems involving exponents by applying various rules, such as the power of a power rule \((a^m)^n = a^{m\times n}\), which simplifies the expression by combining the multiplications.

Base of Logarithm

Logarithms can be tricky, but they essentially undo exponentiation. When dealing with logarithms, it’s essential to understand the base. The base of a logarithm defines the rate at which the logarithmic function increases.
In the expression \(\log_a(b)\), **a** is the base, and it tells us the power to which **a** must be raised to get **b**.

• A common base is 10, as seen in "common logarithms," symbolized typically as \(\log)b\).
• Another significant base is **e** (approximately 2.718), known as the **natural log** and is denoted by \(\ln(b)\).

Understanding the base is crucial, especially when using properties of logarithms like the one used in the original problem, where the expression \(10^{\log(3x^2+2x+1)}\) capitalized on matching the base of the exponent and logarithm. This led to simplified calculations.
In essence, when bases match, they "cancel out," leaving you with the argument of the logarithm, thus making solving such expressions more straightforward.

Polynomial Simplification

Polynomial simplification involves reducing a polynomial to a simpler or more concise form without changing its value. This process uses algebraic techniques to combine like terms and remove complexities.
Consider a polynomial such as \(3x^2 + 2x + 1\). To simplify it involves:

• Ensuring all like terms are combined. Since this already involves different powers of \(x\), no further simplification of terms within itself is necessary here.
• Applying any additional rules where necessary, such as the distributive property \((a(b+c)=ab+ac)\).

When in the context of logarithms or exponents, the simplification often also examines whether the operations (like exponentiation or factoring) allow these complex polynomial expressions to become more manageable.
In the given exercise, once the expression inside the logarithm finds simplification through proper application of properties, you return directly to the polynomial, indicating its simplicity is inherently intact. This helps students recognize how logarithms and polynomial operations intertwine.