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Chapter 8: Problem 57

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

Short Answer

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

Step by step solution

01

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.
02

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 \(4^0 = 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Solve each equation. $$ \log 5-\log 2 x=1 $$

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Solve each equation. If necessary, round to the nearest ten-thousandth. $$ 12^{4-x}=20 $$

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