Problem 60 - Algebra 2

Question

Write each equation in exponential form. $$ \log 10=1 $$

Step-by-Step Solution

Answer

The equation \( \log 10 = 1 \) in exponential form is \( 10^1 = 10 \).

Step 1: Identify the base, exponent and argument from the logarithmic form

From the formula \( \log_b a = n \), the base here is 10 because the base of any log without a base mentioned is 10. The argument of the log (\( a \)) is 10 and the number on the other side of the equation (\( n \)) is 1.

Step 2: Use the formula to convert to exponential form

Putting these values into the formula \( b^n = a \): The base \( b \) is 10, the exponent \( n \) is 1, and \( a \) is 10.

Step 3: Write the equation in exponential form

Replacing \( b \), \( n \), and \( a \) yields the equation in exponential form: \( 10^1 = 10 \).

Key Concepts

Logarithmic Form

Logarithmic form is a way to express the idea of an exponentiation relationship using logarithms. It indicates the power to which a base must be raised to obtain a specific value. A basic logarithmic equation is written as \( \log_b a = n \), where:

• \( b \) is the base of the logarithm.
• \( a \) is the argument or the result of the base being raised to the power of \( n \).
• \( n \) is the exponent or the power to which the base must be raised.

The essence of writing in logarithmic form is to understand and manipulate exponential relationships. This allows us to transform equations so they can be solved in different contexts, such as solving for unknown exponents. Remember, when a logarithm is written without any base, it generally implies a base of 10, known as the common logarithm.

Exponents

Exponents are a shorthand way to denote repeated multiplication of a number by itself. In the expression \( b^n \), the \( b \) is the base and \( n \) is the exponent. This means that \( b \), often referred to as a base number, is multiplied by itself \( n \) times. Here are a few critical points about exponents:

• Any number raised to the power of 1 is the number itself: \( b^1 = b \).
• Any non-zero number raised to the power of 0 is 1: \( b^0 = 1 \).
• Exponents are crucial in expressing large numbers concisely, such as in scientific notation.

Exponents show how many times the base is used as a factor, making it easier to handle large numbers and perform multiple multiplications. Using exponents, complex equations can be efficiently simplified.

Mathematical Conversion

Mathematical conversion involves changing an expression from one form to another while retaining its value. Specifically, converting logarithmic form to exponential form is a common practice for simplifying and solving equations.

• Start by identifying the base \( b \), the argument \( a \), and the output \( n \) from the logarithmic form equation \( \log_b a = n \).
• Use the formula \( b^n = a \) to rewrite the equation in exponential form.
• Perform the calculation if necessary to verify the equivalence.

The conversion from logarithmic to exponential form is essential because it often transforms complicated logarithmic equations into simpler exponential ones. In the context of the given exercise, the equation \( \log 10 = 1 \) was converted to \( 10^1 = 10 \), making the relationship clearer and easier to solve. This process aids in understanding and applying mathematical expressions in various applications.