Problem 25 - Algebra 2

Question

Evaluate each logarithm. $$ \log _{5} 125 $$

Step-by-Step Solution

Answer

The value of \( \log _{5} 125 \) is 3.

Step 1: Understand the given logarithm

The given logarithm is \( \log _{5} 125 \). It's important to remember how logarithms work; a logarithm of base b of a number y is the power to which b must be raised to get y. We write it this way: \( \log _{b} y = x \) if and only if \( b^{x} = y \).

Step 2: Rewrite the logarithm in exponential form

We can rewrite the given logarithm \( \log _{5} 125 \) in exponential form. Using the definition of the logarithm the base 5 raised to the power x equals 125, so we write it as \( 5^{x} = 125 \)

Step 3: Solve for x

Knowing that 5 cubed (5^3) equals 125, we conclude that \( x = 3 \)

Key Concepts

Exponential Form

Exponential form is a way of representing numbers using a base and an exponent. This is particularly useful for showing the relationship between a logarithm and its base and result. In exponential form, a number is expressed as the base raised to a certain power. For example, in the exercise given, the connection between the logarithm and its exponential form is shown clearly.

• The base in the logarithm, which is 5, becomes the base of the exponential equation.
• The result of the logarithm equation 125 becomes the number that the base, 5, is raised to equal.

So, when we write the logarithm \( \log_{5} 125 \), it can be rewritten in exponential form as:\[ 5^{x} = 125 \]This transformation helps us understand logarithms by giving us a clear equation to solve. In this way, we can determine what power the base needs to be raised to in order to achieve the result, fulfilling the original logarithmic equation.

Base of a Logarithm

The base of a logarithm is a fundamental component that dictates the framework of both the logarithm and its corresponding exponential form. A base is essentially the number which is continuously multiplied by itself in the exponential expression.

• For example, in the logarithm \( \log_{5} 125 \), the base is 5.
• The base tells us how many times it is multiplied by itself to result in the value behind the logarithm, which is 125 in this context.

The base is crucial because it provides the scaling factor by which the power is calculated in the logarithm process. Understanding the base helps in rewriting logarithms into exponential form and vice versa, strengthening foundational mathematical skills.Knowing this allows students to simplify and solve logarithmic expressions across various algebraic contexts.

Powers and Exponents

Powers and exponents are key elements in understanding both exponential forms and logarithms. A power indicates how many times a base is multiplied by itself, while an exponent is the little number written above and to the right of the base.

• In the example where we solved \( 5^{x} = 125 \), the exponent is the unknown 'x,' which shows how many times the base (5) must be multiplied by itself.
• Powers help in simplifying expressions and playing with big numbers, which is a step needed particularly in solving logarithms.

In order to solve for the exponent, knowledge of basic powers is useful. Recognizing familiar patterns, such as that 5 raised to the power of 3 equals 125,gives us insight directly into solving the logarithmic equations efficiently. The connection between powers and logarithms is central to navigating algebra problems confidently and correctly, enhancing learning on topics of greater complexity later on.