Question

\(\log _{a} 1\) is equal to (a) 0 (b) 1 (c) \(\infty\) (d) 10

Short answer

The correct answer is (a) 0.

Step-by-step solution

Understanding the properties of logarithms

Recall the property that any number (except 0) raised to the power of 0 gives the result of 1. This can be written as \(a^0 = 1\). We can see here that 'a' is the base and '0' is the exponent.

Apply the property of logarithms to the exercise

We can transfer the above relationship \(a^0 = 1\) into logarithmic form using the definition of a logarithm. The logarithm base 'a' of 1 is equal to the exponent which a must be raised to produce 1. In this case, the exponent is 0. Therefore, the expression \(\log _{a} 1\) equals 0.

Key concepts

Exponents and Logarithms

Exponents and logarithms are fundamental mathematical concepts that relate to each other closely. An exponent refers to the number of times a base number is multiplied by itself. For example, if we have an expression like \( a^n \), "\( a \)" is the base and "\( n \)" is the exponent, meaning we multiply \( a \) by itself \( n \) times. Logarithms, on the other hand, are the inverse operations of exponents. They answer the question: "To what exponent must the base be raised, to produce a particular number?" Written as \( \log_a b = c \), this reads as "the logarithm base \( a \) of \( b \) is \( c \)". This translates to \( a^c = b \).

• Exponents: Multiply the base by itself a certain number of times.
• Logarithms: Find the power the base must be raised to achieve a number.

Understanding this relationship helps simplify expressions and solve problems involving exponential growth or decay.

Logarithm of One

The concept of the logarithm of one is a special property of logarithms. For any positive base \( a \) (except zero), the logarithm base \( a \) of 1 is always 0. This property arises because any base raised to the power of 0 gives 1, as in: \( a^0 = 1 \). So, in logarithmic terms, \( \log_a 1 = 0 \). This property remains consistent regardless of the base, as long as it is positive and not equal to 1. This aspect of logarithms is valuable when dealing with more complex equations, as it simplifies calculations by turning any logarithmic form of 1 to 0.

• Logarithm of 1: Always equals 0 for any positive base.
• Useful property: Simplifies equations and problem-solving.

Base of Logarithm

The base of a logarithm plays a critical role in defining the logarithmic relationship. It is the number that is repeatedly multiplied by itself to reach the exponentiated value. In a logarithmic expression like \( \log_a b \), "\( a \)" is referred to as the base.Having different bases can change the value of logarithms significantly. The most common bases used in mathematics are 10 (common logarithm), \( e \) (natural logarithm), and 2 (binary logarithm), among others. In calculations and scientific notations, selecting the appropriate base is crucial as it impacts the simplification process.

• Common Base Choices: 10, \( e \), and 2.
• Logarithmic Equation: Base helps in understanding the resulting power.

To convert between different bases, certain formulas are applied, such as the change of base formula, which allows transforming logarithms of one base into another for more manageable calculations.