Problem 15 - Algebra 2 and Trigonometry

Question

In \(15-26,\) write each logarithmic equation in exponential form. $$ \log _{10} 100=2 $$

Step-by-Step Solution

Answer

The exponential form is \( 10^2 = 100 \).

Step 1: Identify the Logarithmic Equation Components

The logarithmic equation given is \( \log_{10} 100 = 2 \). Here, the base is 10, the argument is 100, and the logarithm of 100 is equal to 2.

Step 2: Understand the Meaning of Logarithms

The equation \( \log_{10} 100 = 2 \) indicates that 10 raised to some power results in 100. This power is 2, as indicated by the equation; in other words, \( 10^2 = 100 \).

Step 3: Write in Exponential Form

Based on the understanding that a logarithm is the inverse of exponentiation, we convert the logarithmic equation to the exponential equation: \( 10^2 = 100 \). This is the exponential form of the given logarithmic equation.

Key Concepts

Exponential Form

The exponential form is a way of expressing numbers where a base is raised to a power to get another number. In simpler terms, it's like saying "what power should I raise this base number to get this final number?" When you see an equation like \( a^b = c \), this is an exponential form. It consists of:

• The base: the number that's being multiplied by itself.
• The exponent: the power you raise the base to.
• The result: the final number you get once the base is multiplied by itself according to the exponent.

In the original problem, the equation \( \log_{10} 100 = 2 \) translates to \( 10^2 = 100 \) in exponential form. Here, the base is 10, the exponent is 2, and the result is 100. This conversion is crucial as it helps visualize what logarithms really represent.

Base of Logarithm

In a logarithmic equation, the base is the number that is raised to a power to achieve the argument. It's fundamentally important because the base dictates how the rest of the logarithmic expression is interpreted and calculated. Essentially, the base becomes the key part of translating the logarithm into exponential form.In a logarithm like \( \log_{b} x = y \), \( b \) represents the base. It acts as the foundation of exponentiation. The whole concept of logarithms rests on this base, determining how many times it is multiplied by itself to reach the number \( x \). For example in \( \log_{10} 100 = 2 \):

• The base is 10.
• It shows how 10 needs to be multiplied by itself 2 times to become 100.

Understanding the base helps demystify how logarithmic and exponential expressions relate to each other.

Inverse of Exponentiation

Logarithms are considered the inverse of exponentiation, meaning they "undo" what exponentiation does. If exponentiation tells you what number you get when multiplying the base by itself according to the exponent, logarithms tell you the exponent you need to raise the base to, to get that number.Imagine this relationship as a see-saw, where exponentiation and logarithms balance each other out. For instance, if you know \( b^y = x \), then \( \log_{b} x = y \). They both describe the same mathematical relationship, just viewed from different perspectives.In our example, \( 10^2 = 100 \) is the exponential form showing exponentiation, while \( \log_{10} 100 = 2 \) shows the logarithmic, or inverse, view. Recognizing this inverse relationship is vital for converting between logarithmic and exponential forms, making these concepts valuable to understand for solving equations.