Problem 73 - Algebra 2

Question

Write an equivalent exponential equation. $$ \log _{5} 125=3 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Logarithmic Equation

The given equation is \( \log_{5} 125 = 3 \), which means in logarithmic form that the base \( 5 \) raised to the power of \( 3 \) equals \( 125 \).

Step 2: Convert the Logarithmic Equation to an Exponential Equation

To convert the logarithmic equation of the form \( \log_{b}(a) = n \) to its equivalent exponential form, you use \( b^n = a \). In this problem, substitute \( b = 5 \), \( n = 3 \), and \( a = 125 \).

Step 3: Write the Exponential Equation

Using the conversion from Step 2, the exponential form of the equation is \( 5^3 = 125 \), which confirms that the base \( 5 \) raised to the power of \( 3 \) is indeed \( 125 \).

Key Concepts

Logarithms

Logarithms are mathematical expressions that help us solve equations involving powers more easily. They are the inverse operation of exponents. When you have a logarithm like \( \log_{b}(a) = n \), the base \( b \) is the number that needs to be raised to the power \( n \) to get \( a \). In simpler terms, a logarithm answers the question "To what power should we raise this base to get that number?"

• If we say \( \log_{5}(125) = 3 \), it means the power that base \( 5 \) needs to be raised to make \( 125 \) is 3.
• Logarithms make it easier to handle large numbers or solve exponential equations.

By understanding logarithms, you can seamlessly switch between exponential forms and logarithmic forms.

Base and Exponent Concepts

Understanding the base and exponent in mathematical terms is key for solving logarithmic and exponential equations. The base, often denoted as \( b \), is the number that gets multiplied by itself, and the exponent, shown as \( n \), is how many times to multiply the base by itself. Consider the expression \( b^n \):

• \( b \) is the base - it's like the foundation of a building.
• \( n \) is the exponent - it tells you how many floors (multiplications) the building has.

For example, in the expression \( 5^3 \), 5 is the base, and 3 is the exponent, meaning you'll multiply 5 by itself three times: \( 5 \times 5 \times 5 = 125 \). This understanding is essential for converting between logarithmic and exponential forms.

Equivalent Equations

Equivalent equations are different ways of expressing the same mathematical relationship. In the context of logarithms and exponents, they help in understanding and solving equations by converting them into simpler or more familiar forms. When you have an equation like \( \log_{5}(125) = 3 \), you can convert it to an exponential equation, \( 5^3 = 125 \), to make it easier to understand or solve. This conversion doesn't change the value, but it offers a different perspective.

• Logarithmic form: Expresses the equation using a logarithm, which can sometimes simplify solving it.
• Exponential form: The same relationship presented as a multiplication of the base by itself numerous times, easier for some calculations.

By converting between these forms, you solidify your understanding of the relationship and develop flexibility in solving complex problems.