Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-14,\) write each exponential equation in logarithmic form. $$ 5^{3}=125 $$

Step-by-Step Solution

Answer

The logarithmic form is \( \log_5{125} = 3 \).

Step 1: Identify Exponential Equation Components

The given exponential equation is \( 5^3 = 125 \). Here, the base is 5, the exponent is 3, and the result is 125.

Step 2: Apply Logarithmic Form

In the logarithmic form, we swap the position of the base and the result as follows: \(\log_{base}(result) = exponent\). So, the equation becomes \( \log_5{125} = 3 \).

Step 3: Confirm Conversion

Verify that the logarithmic form \( \log_5{125} = 3 \) correctly expresses the relationship that raising 5 to the power of 3 gives 125, confirming the conversion was accurate.

Key Concepts

Exponential Equations

Exponential equations involve a mathematical expression where a constant base is raised to a power or an exponent, resulting in a particular value. In our example, the equation is given as \( 5^3 = 125 \).
This means the base, 5, is multiplied by itself 3 times (5 * 5 * 5) to arrive at the value 125. Exponential equations are widely used in various mathematical and scientific fields for expressing exponential growth or decay.
Such equations serve as the foundation for many real-world applications, such as calculating compound interest, population growth, and radioactive decay. Understanding the components—base, exponent, and result—is crucial for solving any exponential equation.

• Base: The constant number being multiplied.
• Exponent: Indicates how many times the base is used as a factor.
• Result: The value obtained after applying the exponent to the base.

Converting Exponential to Logarithmic

Converting an exponential equation to a logarithmic form is done by rearranging the relationship between the base, exponent, and result. The logarithmic form provides a useful way to work in situations where the exponent is unknown and needs to be solved. For example, the exponential equation \( 5^3 = 125 \) is converted into logarithmic form as \( \log_5{125} = 3 \).
Here, the base of the exponent becomes the base of the logarithm, the result becomes the argument of the logarithm, and the exponent becomes the solution. This conversion is beneficial because logarithmic equations can simplify complex multiplication and division into easier addition and subtraction operations.
The steps for conversion include:

• Identify: Recognize the components—base, exponent, result.
• Swap Positions: Rearrange them into \( \log_{base}(result) = exponent \).
• Verify: Ensure the logarithmic equation correctly expresses the original exponential relationship.

Logarithms in Algebra

Logarithms play a vital role in algebra by offering a method to solve complex equations involving exponents. They help transform multiplicative relationships into additive ones, making calculations simpler.
For instance, when dealing with large numbers or variables, logarithms can reduce exponential growth rates to manageable linear equations. This makes it easier to find unknown exponents.
Here are some key functions of logarithms in algebra:

• Solve for Exponents: When a direct solution is difficult, convert to logarithmic form.
• Simplify Expressions: Turn products into sums using properties of logarithms.
• Change of Base: Use the property \( \log_b{a} = \frac{\log_c{a}}{\log_c{b}} \) to change the base of a logarithm.
• Algebraic Manipulation: Enable the solving of equations involving powers and roots.

Understanding the properties and laws of logarithms is essential in mastering algebra. These properties include product, quotient, and power rule, which allow efficient handling of algebraic problems.