Question

State a value of \(x\) so that \(\log _{3} x\) is a) a positive integer b) a negative integer c) Zero d) a rational number

Short answer

a) x = 3 b) x = 1/3 c) x = 1 d) x = sqrt(3)

Step-by-step solution

Understanding Logarithms

The logarithm \(\text{log}_3 x\) represents the power to which the base 3 must be raised to produce the number x. For example, \(\text{log}_3 27 = 3\) because \({3^3 = 27}\).

- Solve for a Positive Integer

For \(\text{log}_3 x\) to be a positive integer, \(x\) has to be a power of 3 greater than 1. For example, \(3^1 = 3\), so \(\text{log}_3 3 = 1\). Thus, one value for \(x\) is 3.

- Solve for a Negative Integer

For \(\text{log}_3 x\) to be a negative integer, \(x\) has to be a fraction wherein the base 3 is in the denominator. For example, \(3^{-1} = \frac{1}{3}\), so \(\text{log}_3 \frac{1}{3} = -1\). Thus, one value for \(x\) is \(\frac{1}{3}\).

- Solve for Zero

For \(\text{log}_3 x\) to be zero, \(x\) must be equal to 1 since any base raised to the power of 0 is 1. Therefore, \(\text{log}_3 1 = 0\). Thus, \(x = 1\).

- Solve for a Rational Number

For \(\text{log}_3 x\) to be a rational number, \(x\) must be some fraction which can be expressed as \(3^{a/b}\) where a and b are integers. For example, \(x = 3^{1/2} = \text{sqrt}(3)\), so \(\text{log}_3 \text{sqrt}(3) = \frac{1}{2}\). Thus, one value for \(x\) is \(\text{sqrt}(3)\).

Key concepts

positive integers

To find a value of \( x \) such that \( \text{log}_3 x \) is a positive integer, we need to understand how logarithms work with powers of numbers. A logarithm is simply the exponent to which the base must be raised to produce a given number. When we talk about a positive integer as the result of a logarithm, we are essentially saying that \( x \) must be a power of 3 greater than 1. For instance, if \( x \) equals 3, then \( \text{log}_3 3 = 1 \). Since 1 is a positive integer, 3 is a valid value for \( x \). Likewise, \( \text{log}_3 9 = 2 \) and \( \text{log}_3 27 = 3 \) because \( 3^2 = 9 \) and \( 3^3 = 27 \). Note that in general, any value of the form \( 3^n \), where \( n \) is a positive integer, will satisfy the condition.

• Power of 3 greater than 1
• Examples: \( 3, 9, 27 \)

negative integers

When we need \( \text{log}_3 x \) to be a negative integer, we are looking for values of \( x \) that position 3 in a fractional form where the exponent is negative. Specifically, \( x \) must be the reciprocal of a power of 3. For example, if \( x = \frac{1}{3} \), then \( \text{log}_3 \frac{1}{3} = -1 \) because \( 3^{-1} = \frac{1}{3} \). Similarly, \( \text{log}_3 \frac{1}{9} = -2 \) and \( \text{log}_3 \frac{1}{27} = -3 \). Thus, any fraction \( 3^{-n} \), where \( n \) is a positive integer, will yield a negative integer logarithm.

• Reciprocal of a power of 3
• Examples: \( \frac{1}{3}, \frac{1}{9}, \frac{1}{27} \)

zero in logarithms

For the logarithm of a number to be zero, the base raised to the power of zero must equal that number. In simpler terms, since any number raised to the zero power is 1, we set \( \text{log}_3 x = 0 \) and need \( x \) to be 1. Thus, \( \text{log}_3 1 = 0 \) effectively, as \( 3^0 = 1 \).

• Base raised to zero power equals 1
• Example: \( x = 1 \)

rational numbers in logarithms

A rational number can be expressed as a fraction of two integers. For \( \text{log}_3 x \) to be a rational number, \( x \) must be expressible as \( 3^{a/b} \), where \( a \) and \( b \) are integers. One clear example is \( 3^{1/2} \), which equals the square root of 3. Thus, \( \text{log}_3 \text{sqrt}(3) = \frac{1}{2} \). More generally, any number where \( x = 3^{m/n} \), where \( m \) and \( n \) are integers, will work. Other examples include \( 3^{2/3} \) or \( 3^{-1/2} \).

• Expressible as \(3^{a/b}\)
• Examples: \( \text{sqrt}(3), 3^{2/3}, 3^{-1/2} \)