Problem 42 - Algebra 2 and Trigonometry

Question

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ \log _{5} 125 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to evaluate the logarithmic expression \( \log_{5} 125 \), which asks us to find the power to which 5 must be raised to result in 125.

Step 2: Use the Definition of Logarithms

By the definition of logarithms, \( \log_{b} a = c \) means that \( b^c = a \). Here, we set \( \log_{5} 125 = x \), which implies that \( 5^x = 125 \).

Step 3: Recognize 125 as a Power of 5

Recognize that 125 is a power of 5. Specifically, 125 can be expressed as \( 5^3 \) because \( 5 \times 5 \times 5 = 125 \).

Step 4: Solve the Exponential Equation

Since we established that \( 5^x = 5^3 \), we can equate the exponents: \( x = 3 \).

Step 5: Conclude the Result

Thus, the value of \( \log_{5} 125 \) is 3.

Key Concepts

Evaluating Logarithms

Logarithms might seem tricky at first, but they actually represent a simple concept. Evaluating a logarithm like \( \log_{5} 125 \) involves finding out how many times the base number, here 5, must be multiplied by itself to reach another number—in this case, 125.

You start by understanding the basic definition of a logarithm. The expression \( \log_{b} a = c \) signifies "the power \( c \) to which the base \( b \) must be raised to obtain \( a \)." In our example, we set \( \log_{5} 125 = x \) which means finding \( x \) such that \( 5^x = 125 \).

• Think of it visually: How many times would you multiply 5 by itself to get 125? (Answer: 5 multiplied by itself three times equals 125)
• Once this relation is established, solving for \( x \) becomes straightforward.

Understanding this helps to demystify logarithms. They are merely asking "to what power or exponent must we raise a number to get another number?" This is key to recognizing the power of logarithms.

Exponential Equations

Exponential equations, like the one we encountered in the exercise with \( 5^x = 125 \), are equations where variables are found in the exponent. These can often be resolved by expressing both sides of the equation with the same base.

For the problem of \( 5^x = 125 \), we recognized that 125 can be expressed as a power of 5. The trick is identifying recognizable bases common to both sides of the equation:

• Rewrite 125 as \( 5^3 \), which means 125 is achieved by multiplying 5 three times \((5 \times 5 \times 5)\).
• Once we have \( 5^x = 5^3 \), the bases are the same, allowing us to equate the exponents: \( x = 3 \).

This method is very handy for solving exponential equations, especially when numbers are a known power of each other. Knowing this makes working with similar equations much easier.

Properties of Exponents

The properties of exponents are fundamental in solving logarithms and exponential equations. Understanding these properties empowers you to simplify and solve equations seamlessly. Here's a brief overview along with the context of our problem:

• Product of Powers: \( a^m \times a^n = a^{m+n} \) - Tells us when multiplying similar bases, you add the exponents.
• Power of a Power: \((a^m)^n = a^{m \times n}\) - Influences how exponents themselves can multiply.
• Power of a Product: \((ab)^m = a^m \times b^m\) - A power can apply to all terms in a product.

In the exercise, identifying that \( 125 = 5^3 \) uses a form of the power of a number—the essence of working with exponents. Recognizing numbers as powers and using these properties simplifies the expression, making it more manageable and revealing the value of \( x \) easily.
Always remember: exponents are tools that can reorder and simplify equations, helping to solve them effectively.