Problem 57 - Algebra 2

Question

Write each equation in exponential form. $$ \log _{4} 1=0 $$

Step-by-Step Solution

Answer

The exponential form of the given logarithmic equation \(\log _{4} 1=0\) is \(4^0 = 1\).

Step 1: Understanding the log notation

Look at the equation \(\log _{4} 1=0\). The number '4' is the base. The result of the logarithmic operation is '0' and '1' is the number for which we're finding the logarithm.

Step 2: Convert the log equation to exponential form

The log equation can be converted into exponential form based on the principle that if \(\log _{b} y = x\), then \(b^x = y\). Replacing 'b' with '4', 'x' with '0' and 'y' with '1', we get \(4^0 = 1\).

Key Concepts

Logarithms

A logarithm is a mathematical concept that helps us understand the power to which a number, called the base, must be raised to obtain another number. In simpler terms, it answers the question: "To what power should we raise the base to get a certain number?" For example, in the equation \( \log_4 1 = 0 \), we are trying to determine the power we need to raise 4 to in order to obtain 1.
\(\log_4 1\) translates to asking which power of 4 gives us 1, and the answer is 0, because any number raised to the power of 0 equals 1.
Logarithms are widely used because they make it easier to work with powers and exponents, especially when dealing with very large or very small numbers.

• The base of the logarithm tells us which number is being repeatedly multiplied in the context of the problem.
• The result, in this case 0, tells us how many times the base is multiplied by itself to reach the given number.

Base of a Logarithm

The base of a logarithm is a crucial component in understanding and solving logarithmic equations. It signifies the number that is repeatedly multiplied to approach the result. When we see a logarithm such as \(\log_4\), the base is 4.
In the expression \(\log_b y = x\), the base \(b\) determines the rate at which we multiply. The property of a logarithm is such that if we change the base, it changes the entire behavior of the equation.
The exponential form \(b^x = y\) is derived from the logarithm \(\log_b y = x\) and reiterates this behavior. A deep understanding of the base helps in converting between logarithmic and exponential forms efficiently and correctly.

Exponential Equations

Exponential equations involve numbers that grow by being raised to a power, or exponents. These kinds of equations are a direct opposite to logarithms, which decompose or break down a number to find the exponent.
In converting a logarithmic expression to exponential form, the logic is to rearrange the terms so that the base, exponent, and result are clearly defined. With our example, the logarithm \(\log_4 1 = 0\) converts to the exponential equation \(4^0 = 1\).
This transformation is essential, as it helps in visualizing the power relationship between the base and the result. It simplifies solving equations where the unknown is an exponent.

• Understanding exponential forms aids in simplifying complex calculations.
• They are used in various applications such as computing compound interest and growth rates in populations.