Problem 23 - Algebra A Combined Function

Question

Write each as a logarithmic equation. $$ 10^{-1}=\frac{1}{10} $$

Step-by-Step Solution

Answer

\( \log_{10}(\frac{1}{10}) = -1 \)

Step 1: Identify the Base, Exponent, and Result

In the given exponential equation, the base is 10, the exponent is -1, and the result is \( \frac{1}{10} \).

Step 2: Convert the Exponential Form into a Logarithmic Form

To convert the exponential equation \( 10^{-1} = \frac{1}{10} \) into a logarithmic form, use the formula: \( \log_{\text{base}}(\text{result}) = \text{exponent} \).

Step 3: Apply the Formula

Using the formula determined in Step 2, substitute the values: \( \log_{10}(\frac{1}{10}) = -1 \).

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a constant base is raised to a variable exponent. In this context, the equation is written in the form \(a^b = c\), where \(a\) is the base, \(b\) is the exponent, and \(c\) is the result. Exponential equations are fundamental as they describe exponential growth or decay, such as population growth or radioactive decay.
In the given exercise, the exponential equation is \(10^{-1} = \frac{1}{10}\). This equation illustrates how changing the exponent affects the base and the resulting calculation. Understanding these kinds of equations is crucial in fields like finance and science, where they model real-life exponential processes.
Analyzing exponential equations helps in solving complex mathematical problems by transforming them into simpler forms like logarithms, which are more manageable.

Base and Exponent

The base and the exponent are two key components of exponential equations. The base is the number that is being raised to the power of the exponent. It serves as the foundational value in an exponential expression.
For example, in the expression \(10^{-1}\), the base is 10. This is the number that will be multiplied by itself a specified number of times, indicated by the exponent.
The exponent is the number that determines how many times the base is used as a factor. In the equation \(10^{-1} = \frac{1}{10}\), the exponent \(-1\) signifies that the base 10 should be taken to the inverse power, resulting in a fractional form: \(\frac{1}{10}\).
Exponents can be positive or negative. Positive exponents explain multiplication, while negative exponents reflect division or the reciprocal of the base raised to the positive exponent. Mastering this concept is essential for simplifying and solving exponential equations efficiently.

Converting Forms

The process of converting exponential equations to logarithmic form involves reinterpreting the relationship between the base, exponent, and result. This technique is a fundamental skill that enables mathematicians to transition between different mathematical expressions.
To convert an exponential equation like \(a^b = c\) into a logarithmic form, one uses the formula: \(\log_{a}(c) = b\). It effectively expresses the exponent as a log of the result with the given base.
For instance, the original exponential equation \(10^{-1} = \frac{1}{10}\) is converted into a logarithmic equation through this process, resulting in \(\log_{10}(\frac{1}{10}) = -1\).
This conversion is especially useful because logarithms can simplify the process of solving equations by allowing operations such as multiplication and division to be transformed into addition and subtraction.

Logarithmic Form

Logarithmic form is a way to express exponential relationships using logarithms, which are the inverses of exponentials. It provides a powerful tool for tackling complex mathematical problems by transforming exponential relationships into a linear format.
In the logarithmic form, \(\log_{a}(c) = b\), \(a\) is the base, \(c\) is the result, and \(b\) is the exponent. This form demonstrates how the result, when expressed as a power of the base, equals the exponent.
In the given example, the conversion from \(10^{-1} = \frac{1}{10}\) leads to the logarithmic form \(\log_{10}(\frac{1}{10}) = -1\). This equation shows that \(10\) raised to what power equals \(\frac{1}{10}\)? The answer is \(-1\).
By using logarithmic form, problems involving powers and roots become more approachable, allowing for easier calculation and understanding of the underlying mathematical conceptual framework.