Question

The least positive integral value of the expression \(\frac{1}{2} \log _{10} \mathrm{~m}-\log _{\mathrm{m}-2} 10\) is (1) 0 (2) 1 (3) 2 (4) \(-1\)

Short answer

Answer: (2) 1

Step-by-step solution

Convert the expression

Given the expression \(\frac{1}{2} \log _{10} m - \log _{m-2} 10\), start by converting this expression to log base m-2. Use the change of base formula: \(\log _{b} a = \frac{\log_{c} a}{\log_{c} b}\), where a is the number to be converted, b is the original base, and c is the new base. In this case, let's change \(\frac{1}{2} \log _{10} m\) to the base of m-2. We get \(\frac{\frac{1}{2} \log _{m-2} m}{\log _{m-2} 10} - \log _{m-2} 10\)

Combine the logs

Now we can combine the logarithms using the rule: \(\frac{\log _{b} a}{\log _{b} c} - \log _{b} c = \log _{b} \frac{a}{c^2}\). Our expression becomes: \(\log _{m-2} \frac{m}{(10)^2}\)

Find the least positive integral value

To find the least positive integral value of the expression, we should set the expression equal to the least positive integer (which is 1) and solve for m. \(\log _{m-2} \frac{m}{(10)^2} = 1\)

Convert back to exponential form

To solve for m, we need to convert the logarithm back to exponential form. We can do this using the rule: \(\log _{b} a = c \Rightarrow b^c = a\). Here, our equation becomes: \((m-2)^1 = \frac{m}{(10)^2}\)

Solve the equation for m

Now we have a simpler equation to solve for m: \(m-2 = \frac{m}{100}\) Multiply both sides by 100 to eliminate the fraction: \(100m-200 = m\) Combine like terms and solve for m: \(99m = 200\) \(m=\frac{200}{99}\)

Determine the least positive integral value

The value of m we found is not an integer, so we need to look for the least positive integral value for the expression. Here are the steps we'll take to determine that: 1. Check if \(\frac{200}{99}\) is less than 2 (since 2 is the next positive integer after 1). 2. Evaluate the expression \(\log _{m-2} \frac{m}{(10)^2}\) using \(m=3\), where \(m-2 = 1\), which gives an integer solution of 1. 3. Check the other options by plugging in the given choices for the expression and choosing the least integer value. Plugging in \(m=3\) into our expression results in the least positive integral value of 1. So, the answer is (2) 1.

Key concepts

Change of Base Formula

The change of base formula is an essential concept in the realm of logarithms, playing a crucial role in simplifying logarithmic expressions and solving equations. It allows us to rewrite a logarithm in terms of a base of our choosing, which is very helpful when we do not have a calculator that can handle logs of any base, or when we want to simplify the expression for further analysis.

For any positive numbers a, b, and c, where a and b are not equal to 1, the change of the base formula is expressed as: \
\[ \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \]
This formula shows that a logarithm with base b can be converted into a fraction of two logarithms with a new base c. In the context of our example from the textbook solution, using the change of base formula enabled us to change \( \frac{1}{2} \log _{10} m \) to log base m-2, which prepared the stage for further simplification steps.

Logarithm Properties

Understanding the properties of logarithms can greatly simplify the process of working with log expressions and solve logarithmic equations effectively. These properties are rules that apply to logarithms, allowing us to manipulate and combine them in various ways.

Here are some fundamental properties of logarithms:

• The Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \), meaning the log of a product is the sum of the logs.
• The Quotient Rule: \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \), which states that the log of a quotient is the difference of the logs.
• The Power Rule: \( \log_b(m^n) = n \cdot \log_b(m) \), which means we can take the exponent and make it a coefficient.

By applying these properties, especially the Quotient Rule, we could combine the logarithmic terms in the given textbook expression into a single logarithm, setting the stage for finding the solution.

Solving Logarithmic Equations

Solving logarithmic equations typically involves a series of strategic steps to isolate the logarithmic term and then transform the equation into an exponential form, which often makes it easier to solve. As demonstrated in the solution steps from the exercise, this approach can unveil the value of an unknown that satisfies the equation.

Here is a general approach to solving logarithmic equations:

• First, apply logarithm properties to combine or simplify the logarithm terms.
• Second, if the equation is in the form \( \log_b(x) = c \), use the definition of a logarithm to convert it to its exponential form, which is \( b^c = x \).
• Finally, solve the resulting exponential equation for the unknown variable.

By converting the logarithmic expression to an exponential form in our textbook example, we obtained a straightforward algebraic equation. Solving this equation yielded the value of m, which led to us testing integral values to find the least positive integral value satisfying the initial logarithmic expression.