Problem 256 - Calculus Volume 1

Question

Write the equation in equivalent logarithmic form. \(10^{2}=100\)

Step-by-Step Solution

Answer

The equivalent logarithmic form is \(\log_{10} 100 = 2\).

Step 1: Identify the Exponential Form

The given equation is in exponential form: \(10^2 = 100\). In this format, \(10\) is the base, \(2\) is the exponent, and \(100\) is the result.

Step 2: Recap the Logarithmic Form Definition

The logarithmic form \(\log_b a = c\) is defined such that the base \(b\) raised to the power of \(c\) gives the number \(a\). Compare this with the exponential form \(b^c = a\).

Step 3: Apply the Logarithmic Form

Using the logarithmic form definition, write the equivalent equation for \(10^2 = 100\). The base \(10\) and result \(100\) correspond to \(b\) and \(a\) respectively. The equation is \(\log_{10} 100 = 2\).

Step 4: Verify the Transformation

Confirm that the conversion is correct by checking: If \(\log_{10} 100 = 2\), then \(10^2 = 100\). This verification also shows that the logarithmic form is equivalent to the given exponential expression.

Key Concepts

Exponential Form

Exponential form is a way to represent numbers by using powers. It expresses a number as a base raised to a certain power, known as the exponent. For instance, in the expression \(10^2 = 100\), the base is \(10\), the exponent is \(2\) and the result of this operation is \(100\). Whenever you see a number in this style, it is indicating repeated multiplication. Here, \(10\) is multiplied by itself once to achieve \(100\).

If we break it down, exponentiation is:

• Base: The number that will be multiplied.
• Exponent: Tells us how many times to multiply the base.
• Result: The final outcome after performing the multiplication.

This not only makes it easier to write large numbers but also simplifies calculations. Understanding exponential form is essential for transitioning to logarithmic form, as they are interconnected mathematical concepts.

Logarithmic Form

The logarithmic form is another way to express equations which are originally in exponential form. It makes it easier to solve for unknowns, especially when dealing with large numbers or complex calculations. The logarithmic equation format is \(\log_b a = c\). This means "logarithm base \(b\) of \(a\) equals \(c\)," which is directly connected to the exponential form \(b^c = a\).

When you encounter an exponential equation, such as \(10^2 = 100\), you can switch it to its logarithmic form \(\log_{10} 100 = 2\). This transformation helps to isolate different components of the equation.

• Base \(b\): Remains the same in both forms.
• Exponent \(c\) in the exponential form becomes the answer or result in the logarithmic form.
• The result \(a\) in the exponential equation is the number for which we find the log.

By converting from exponential to logarithmic, one can solve equations involving exponents with more ease, as logarithms turn multiplication into addition, which is generally more straightforward to handle.

Base and Exponent

In both exponential and logarithmic forms, there are key components: the base and the exponent. Understanding these is crucial to mastering the concepts.

• Base: This is the number that gets multiplied in exponential expressions. It remains constant between exponential and logarithmic conversions.
• Exponent: This tells how many times the base is used in a multiplication sequence. It becomes the logarithmic expression's result when switching forms.

For example, in the equation \(10^2 = 100\), \(10\) is the base, and \(2\) is the exponent. In the logarithmic form \(\log_{10} 100 = 2\), \(10\) remains the base, while \(2\) is what the calculation equates to, essentially identifying how we arrived at \(100\) using \(10\).

These components help decipher the operation needed to achieve a result. For instance, larger bases or exponents indicate larger results or more repeated multiplication steps. The base and exponent are fundamental to understanding how equations can be manipulated and interpreted in different mathematical scenarios.