Problem 3 - Intermediate Algebra

Question

For Problems \(1-10\), write each of the following in logarithmic form. For example, \(2^{3}=8\) becomes \(\log _{2} 8=3\) in logarithmic form. $$ 5^{3}=125 $$

Step-by-Step Solution

Answer

\(\log_5 125 = 3\).

Step 1: Identify the base, exponent, and result

In the equation \(5^3 = 125\), identify the base, exponent, and result. Here, the base is 5, the exponent is 3, and the result is 125.

Step 2: Write the logarithmic form of the equation

To convert the equation into logarithmic form, we use the definition: if \(b^e = a\), then \(\log_b a = e\). In this case, with \(b = 5\), \(a = 125\), and \(e = 3\), the logarithmic form becomes \(\log_5 125 = 3\).

Key Concepts

Exponents

Exponents are fundamental to understanding logarithms. When you see something like \(5^3 = 125\), the number 5 is called the "base," and the number 3 is the "exponent." The exponent tells us how many times the base is multiplied by itself. So in this case, \(5^3\) means \(5 \times 5 \times 5\), which equals 125.
Understanding exponents is crucial because they allow us to express large numbers in a compact form. They're also essential in scientific notation, growth models, and computer algorithms. When you work with exponents, remember:

• The base is the number being multiplied.
• The exponent shows how many times the base is used in multiplication.
• The entire expression, \(5^3\), is called a "power."

By grasping these basics, you'll be better prepared to delve into logarithms, which are effectively the "opposite" of exponents.

Bases in Logarithms

Bases in logarithms work much like bases in exponents, and they're vital for understanding how logarithms are formed. When you see a logarithmic expression, like \(\log_5 125 = 3\), the base \(b\) here is 5. This indicates the power to which the base must be raised to get the given result, in this case, 125.
In simpler terms:

• The base is the number you repeatedly multiply in exponentiation.
• In a logarithm, the base tells you which number is being transformed by powers.
• Just like in exponents, the base in logarithms needs to be positive and not equal to 1.

Remember that the choice of base can change how the logarithm appears but does not change the inherent essence of the logarithm. The logarithm essentially counts how many times you multiply the base by itself to reach a specific number.

Conversion to Logarithms

Conversion to logarithms is a key skill that bridges the gap between exponents and logarithms. When converting an exponential equation like \(5^3 = 125\) into a logarithmic form, we follow a specific rule: if \(b^e = a\), then it translates into \(\log_b a = e\).
Let's break it down:

• Identify the base, which is 5 in this case.
• Determine the exponent, which is 3.
• Find the result of the exponentiated base, which is 125.

Using this information, you structure the logarithmic form by placing the result 125 after \(\log_5\) (logarithm with base 5), and set it equal to the exponent, 3. Thus, you write it as \(\log_5 125 = 3\).
By mastering this conversion, you unlock the ability to toggle between exponential and logarithmic forms easily, which is a powerful tool in simplifying complex calculations and solving equations.