Problem 23 - Algebra 2

Question

Write each equation in exponential form. \(\log _{5} 125=3\)

Step-by-Step Solution

Answer

The equation in exponential form is \(5^3 = 125\).

Step 1: Understanding the Equation

The given equation is in logarithmic form: \( \log_{5} 125 = 3 \). This means 125 is the result when base 5 is raised to the power of 3.

Step 2: Rewriting in Exponential Form

To convert the logarithmic equation \( \log_{5} 125 = 3 \) into exponential form, we need to use the definition of a logarithm: \( \log_{b} a = c \) is equivalent to \( b^c = a \).

Step 3: Apply the Definition

Using the definition, \( \log_{5} 125 = 3 \) converts to exponential form as \( 5^3 = 125 \).

Step 4: Verification

To ensure the correctness of the conversion, verify by calculating: \( 5^3 = 5 \times 5 \times 5 = 125 \). The calculation is accurate, confirming the exponential form is correct.

Key Concepts

Logarithmic form

Logarithmic form is a way of expressing equations where a number is represented as a base raised to a power. In the context of logarithms, the base indicates the number that is multiplied by itself a certain number of times. The formula \( \log_{b} a = c \) can be broken down as follows:

• \( b \) is the base
• \( a \) is the result
• \( c \) is the exponent or the number of times the base is multiplied by itself to achieve \( a \)

For example, \( \log_{5} 125 = 3 \) represents the exponential equation where 5, the base, is taken to the power of 3 to get 125. **Understanding logarithmic form is crucial because it allows you to solve for unknown exponents by converting them into a more calculable form.** This is especially helpful in situations involving growth and decay processes, such as compound interest and radioactive decay.

Logarithms

Logarithms are the inverse operations of exponentiation, just like subtraction is the inverse of addition. They provide a way to find out the power to which a base number should be raised to obtain another number. This concept is especially useful in many mathematical fields, such as calculus, algebra, and real-world applications like sound intensity and pH levels.**Here are some useful properties of logarithms:**

• Product Property: \( \log_{b}(xy) = \log_{b}(x) + \log_{b}(y) \)
• Quotient Property: \( \log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y) \)
• Power Property: \( \log_{b}(x^y) = y\log_{b}(x) \)

These properties make it easier to manipulate and simplify logarithmic expressions during problem solving.
Logarithms are particularly handy when dealing with very large or very small numbers, by reducing multiplication operations into more manageable addition and subtraction tasks.

Exponents

Exponents are a fundamental part of mathematics, representing repeated multiplication of a number by itself. When a number is raised to the power of an exponent, it means multiplying the number (base) by itself as many times as indicated by the exponent. For instance, \( 5^3 \) means \( 5 \times 5 \times 5 \), which equals 125.Exponential notation is a concise way to express very large or very small numbers.
Here are some important rules:

• Multiplication of Same Base: \( a^m \times a^n = a^{m+n} \)
• Division of Same Base: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power: \( (a^m)^n = a^{mn} \)
• Zero Exponent Rule: \( a^0 = 1 \), assuming \( a eq 0 \)

Understanding how to use exponents is essential not only for solving equations, but also for understanding scientific notation and financial calculations involving compound interest.