Problem 33 - Algebra 2 and Trigonometry

Question

Expand each expression using the properties of logarithms. \(\log _{10}(x+1)^{2}\)

Step-by-Step Solution

Answer

\(2 \times \log_{10}(x+1)\)

Step 1: Identify the Expression to Expand

The given expression is \(\log_{10}(x+1)^2\). We need to expand this expression using the properties of logarithms.

Step 2: Apply the Power Property of Logarithms

The power property of logarithms states that \( \log_b(M^n) = n \times \log_b(M) \). Apply this property to obtain: \( \log_{10}((x+1)^2) = 2 \times \log_{10}(x+1) \).

Key Concepts

Power Property of Logarithms

The power property of logarithms is a handy tool in the toolkit of anyone working with logarithmic expressions. This property simplifies dealing with powers within a logarithm.
Here's the property clearly stated:

• If you have an expression in the form of a logarithm wherein the argument is raised to a power, \(M^n\), then you express it as: \(log_b(M^n) = n imes log_b(M)\),
• This means you can "take down" the exponent and move it in front of the logarithm as a multiplier.

Let's use a simple example to illustrate this:
Consider \( log_2(8^3) \). Using the power property, you can convert this to \( 3 imes log_2(8) \).
This makes solving equations simpler, especially when dealing with large exponents, as it allows the problem to be broken down into manageable parts.

Logarithms

Logarithms are an essential concept in mathematics, especially when dealing with exponential and multiplicative relationships.
Essentially, a logarithm answers the question: "To what exponent must the base be raised to yield a particular number?" For instance, \( log_2(8) = 3 \) because \( 2^3 = 8 \).
The anatomy of a logarithm involves three components:

• Base (b), which is the number you'll repeatedly multiply;
• The Argument (M), which is the result you'll get after multiplying;
• The Result (n), or the exponent to which the base must be raised in order to equal the argument.

Logarithms help transform multiplicative problems into additive ones, much like how the power property simplifies problem-solving. For example, knowing \( log_{10}(1000) = 3 \) allows for easy manipulation in scientific computations.

Logarithmic Expressions

Logarithmic expressions often occur in mathematics, science, and engineering, typically as part of solving equations or modeling real-world phenomena.
The magic of logarithmic expressions lies in their ability to simplify complex multiplication and exponentiation. This is achieved through the key properties of logarithms: product, quotient, and power.When you're asked to "expand a logarithmic expression," it means transforming it in a way that is easier to work with—often by converting a single logarithm with a complex argument into a sum or difference of logarithms.
For example, to expand \( log_{10}((x+1)^2) \), use the power property of logarithms, turning it into \( 2 imes log_{10}(x+1) \).Logarithms can initially seem daunting, but understanding each part allows you to tackle a range of problems.
With practice, you'll notice their applications spanning chemistry, physics, and finance, among others, offering a toolset as powerful as the exponential phenomena they describe.