Question

The value of \(\log _{40} 5\) lies between (1) \(\frac{1}{3}\) and \(\frac{1}{2}\) (2) \(\frac{1}{4}\) and \(\frac{1}{3}\) (3) \(\frac{1}{2}\) and 1 (4) None of these

Short answer

Answer: (3) \(\frac{1}{2}\) and 1

Step-by-step solution

Express 5 and 40 as powers of the same base

We can rewrite 5 and 40 as powers of the same base, 2: 5 = \(2^{2.322}\) and 40 = \(2^{3.322}\). (We use decimals for simplicity)

Use the change of base formula

We will use the change of base formula to switch to base 2 logarithms: \(\log_{40} 5\) = \(\frac{\log_{2}5}{\log_{2}40}\) = \(\frac{\log_{2}2^{2.322}}{\log_{2}2^{3.322}}\)

Simplify the expression

Using logarithm properties, we can simplify the expression as follows: \(\frac{\log_{2}2^{2.322}}{\log_{2}2^{3.322}}\) = \(\frac{2.322}{3.322}\) = \(0.6981\)

Compare with given ranges

We have found the value of \(\log_{40} 5\) to be approximately 0.6981. We will now compare this value to the given ranges: (1) \(\frac{1}{3}\) and \(\frac{1}{2}\) -> \(0.333\) and \(0.500\) (2) \(\frac{1}{4}\) and \(\frac{1}{3}\) -> \(0.250\) and \(0.333\) (3) \(\frac{1}{2}\) and 1 -> \(0.500\) and \(1.000\) (4) None of these Since 0.6981 lies between 0.500 and 1.000, the correct choice is option (3).

Key concepts

Change of Base Formula

Logarithms, in essence, are the inverse operations of exponentiation. They are used to find the power to which a number must be raised to produce a given number. One crucial aspect of logarithms that students frequently encounter is the change of base formula. This formula allows converting a logarithm with any base to a logarithm with a base of your choice, usually to simplify the calculation.
In mathematical terms, the change of base formula is represented as:
\[\begin{equation}\log_b a = \frac{\log_c a}{\log_c b}\end{equation}\]
Where \(b\) is the base we are converting from, \(c\) is the new base we are converting to, and \(a\) is the argument of the logarithm. The beauty of this formula is that it lets you work in bases that are more common, like 10 or e (the base of natural logarithms), when calculators typically have these functions readily available. This technique was exemplified in the provided exercise where a logarithm to the base 40 was converted into a logarithm to the base 2 to make the arithmetic more manageable.
It's also important to note that the choice of base \(c\) in the formula doesn't affect the value of the expression, as long as the base \(c\) is the same in both the numerator and the denominator. This property is what ensures that the identity holds true and the value computed remains consistent regardless of the base used.

Properties of Logarithms

Understanding the properties of logarithms is critical for solving a wide range of mathematical problems. These properties are based on the fundamental nature of logarithms as exponents. The most commonly used properties that facilitate the manipulation and simplification of logarithmic expressions include:

• The Product Rule: \(\log_b(mn) = \log_b(m) + \log_b(n)\), which says that the logarithm of a product is the sum of the logarithms of the individual factors.
• The Quotient Rule: \(\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)\), telling us that the logarithm of a quotient is the difference of the logarithms.
• The Power Rule: \(\log_b(m^n) = n\log_b(m)\), which means that the logarithm of a number raised to an exponent is the exponent times the logarithm of the number.

In the given exercise, the power rule is applied to simplify the logarithmic expression. When the log argument is a power, such as \(2^{2.322}\), it simplifies to the exponent times log of the base, \(2.322 \log_{2}2\), and since \(\log_{b}b = 1\) for any base \(b\), it further simplifies down to just the exponent, 2.322. Familiarizing oneself with these properties not only eases computations but also provides deeper understanding and flexibility in manipulating logarithms for equations and inequalities.

Solving Logarithmic Equations

Solving logarithmic equations can seem intimidating, but armed with a few strategies, it becomes quite manageable. Most often, these strategies involve transforming the logarithmic equation into an exponential one, as this can make the solution more obvious. There are also times we need to apply logarithmic properties to combine or simplify terms before solving.
A systematic approach to solving logarithmic equations may include these steps:

• Isolate the logarithmic term if possible.
• Apply logarithmic properties to simplify the equation.
• Convert the logarithmic equation into an exponential form to solve for the variable.
• Check the solutions in the original equation to avoid extraneous solutions, as not all solutions may satisfy the initial logarithmic equation.

For more complex equations involving multiple logarithmic terms or different bases, additional steps like using the change of base formula, as seen in our exercise, or combining logs into a single logarithmic expression may be required. The key is to recognize that logarithmic equations are solvable through systematic application of properties and rules specific to logarithms. Careful manipulation and comprehension of these rules will allow the successful determination of the unknown variable.