Problem 15 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3.\) $$5^{3}=125$$

Step-by-Step Solution

Answer

The logarithmic form of the given exponential equation \(5^{3}=125\) is \(\log _{5} 125=3\).

Step 1: Identify the base in exponential form

Identify the base \(b\) of the exponential equation. In the given equation \(5^{3}=125\), the base \(b\) is 5.

Step 2: Identify the exponent in exponential form

Identify the exponent \(y\) of the exponential equation. In the given equation \(5^{3}=125\), the exponent \(y\) is 3.

Step 3: Identify the result in exponential form

identify the result \(x\) of the exponential equation. In the given equation \(5^{3}=125\), the result \(x\) is 125.

Step 4: Construct the Logarithmic form

The logarithmic form will be \(\log _{b} x=y\). Substituting the values of \(b\), \(x\) and \(y\) we will get \(\log _{5} 125=3\).

Key Concepts

Exponential Equation

An exponential equation is a type of equation where a variable appears in the exponent. For example, in the expression \( 5^3 = 125 \), "5" is raised to the power of "3". This equation essentially tells us that when 5 is multiplied by itself twice (for a total of three times), it results in 125.An exponential equation consists generally of:

• An exponent: the power to which the base is raised.
• A base: the number that is being multiplied by itself.
• A result: the final value after the multiplication process.

The beauty of exponential equations lies in their simplicity and power to represent very large numbers using just a few components. Understanding exponential equations is foundational in algebra and higher mathematics.

Base of Exponential Form

In an exponential form, the base is the number that is being repeatedly multiplied. It is the "foundation" of the exponential expression. In the equation \( 5^3 = 125 \), the base is 5.The base is significant because:

• It determines the growth rate of the exponential equation.
• It is the number you start with and repeatedly multiply by itself.

Understanding the base is crucial for manipulating and solving exponential equations. Once you know the base, you can start to comprehend how the whole equation comes together.

Exponent

The exponent in an exponential equation tells us how many times the base is used as a factor in multiplication. For instance, in the equation \( 5^3 = 125 \), the exponent is 3.Exponents are important because:

• They indicate the degree of repeated multiplication.
• The higher the exponent, the larger the result, assuming the base is greater than 1.

This simple number elegantly expands the scope of problems you can solve, allowing you to handle enormous numbers effortlessly in both theory and practice.

Conversion between Exponential and Logarithmic Forms

Converting between exponential and logarithmic forms is a key skill in mathematics that often simplifies complex problems. The conversion involves rewriting an exponential equation as a logarithm.For the equation \( 5^3 = 125 \), the logarithmic form is \( \log_{5} 125 = 3 \). Here is how conversion works:

• The base of the exponential equation becomes the base of the logarithm.
• The result of the exponential equation becomes the argument of the logarithm.
• The exponent becomes the answer to the logarithm.

This conversion is helpful to simplify calculations and solving equations where the unknown is in the exponent. It bridges the gap between exponential and logarithmic thinking, providing a different perspective that is particularly useful in algebra, calculus, and real-world applications.