Problem 6 - 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. $$ 10^{1}=10 $$

Step-by-Step Solution

Answer

\(\log_{10}(10) = 1\).

Step 1: Identify the Base, Exponent, and Result

In the expression given, which is \(10^1 = 10\), the base \(b\) is 10, the exponent \(e\) is 1, and the result \(R\) is 10. These are the elements that will be used to write the expression in logarithmic form.

Step 2: Apply Logarithmic Form Definition

The logarithmic form of an exponential expression \(b^e = R\) is written as \(\log_b(R) = e\). Using this definition, we substitute the identified base, result, and exponent into the logarithmic form.

Step 3: Substitute the Values

Now, substitute the base \(b = 10\), the result \(R = 10\), and the exponent \(e = 1\) into the formula \(\log_b(R) = e\). This gives us: \(\log_{10}(10) = 1\). This is the logarithmic form of the given exponential equation.

Key Concepts

Exponential Expressions

Exponential expressions are mathematical statements where a number, known as the base, is raised to the power of an exponent. These expressions help represent repeated multiplication of a base. For example, in the expression \(2^3\), the base \(2\) is multiplied by itself three times, resulting in 8.

Recognizing and understanding exponential expressions is key to many areas of mathematics, including growth calculations and scientific notations. Such expressions are often rewritten or transformed into logarithmic form, especially when solving problems involving growth rates or scales.

• **Base**: The number being multiplied.
• **Exponent**: Indicates how many times the base is used as a factor.
• **Result**: The outcome of the multiplication process.

These components work together to represent complex calculations succinctly.

Base and Exponent

Every exponential expression is made up of two crucial parts: the base and the exponent. The base in an expression, like \(10^1\), is the number that gets multiplied. The exponent signifies how many times the base is used in the multiplication.

In our example, \(10^1 = 10\):

• **Base (\(b\))**: 10 - the number being repeatedly multiplied.
• **Exponent (\(e\))**: 1 - the number of times the base is used.
• **Result (\(R\))**: 10 - the value obtained after the multiplication operation.

Understanding bases and exponents allows us to efficiently convey large numbers and complex operations in a compact form. This understanding is essential for converting between exponential and logarithmic forms.

Logarithmic Equations

Logarithmic equations provide a method to solve for an unknown exponent in exponential expressions, transforming them into a more manageable form. The conversion process involves rewriting the exponential expression \(b^e = R\) as a logarithm: \(\log_b(R) = e\).

For example, the expression \(10^1 = 10\) is rewritten as \(\log_{10}(10) = 1\). Here:

• **Base (\(b\))**: 10 - remains the same in both forms.
• **Result (\(R\))**: 10 - the correct value which the base raised to the exponent equals.
• **Exponent (\(e\))**: 1 - solved and expressed by the logarithm.

This logarithmic form helps in determining the power to which the base must be raised to yield a particular result. It's especially handy in reversing growth problems and solving equations involving exponential growth and decay.