Question

If \(\mathrm{p}=\log _{2 \mathrm{a}} \mathrm{a}, \mathrm{q}=\log _{3 \mathrm{~s}} 2 \mathrm{a}\) and \(\mathrm{r}=\log _{4 a} 3 \mathrm{a}\), so then \(\mathrm{qr}(2-\mathrm{p})=\) (1) 1 (2) 0 (3) 2 (4) 3

Short answer

Answer: The value of $$qr(2-p)$$ is 2.

Step-by-step solution

Understand the given expressions

We are given: p = $$\log_{2a}a$$ q = $$\log_{3a}2a$$ r = $$\log_{4a}3a$$ Our task is to find the value of $$qr(2-p)$$.

Evaluate p

Since p = $$\log_{2a}a$$, let's use the change of base formula for logarithms: $$\log_{b}x = \frac{\log{x}}{\log{b}}$$ So, p = $$\frac{\log{a}}{\log{(2a)}}$$

Evaluate q

Similarly, for q = $$\log_{3a}2a$$, let's use the change of base formula for logarithms: So, q = $$\frac{\log{(2a)}}{\log{(3a)}}$$

Evaluate r

For r = $$\log_{4a}3a$$, let's use the change of base formula for logarithms: So, r = $$\frac{\log{(3a)}}{\log{(4a)}}$$

Calculate qr(2 - p)

Now, let's substitute p, q, and r into the given expression and simplify: qr(2 - p) = $$\left(\frac{\log{(2a)}}{\log{(3a)}}\right)\left(\frac{\log{(3a)}}{\log{(4a)}}\right)(2 - \frac{\log{a}}{\log{(2a)}})$$ After simplification, we get: qr(2-p) = $$2 - \frac{\log{a}}{\log{(2a)}}$$

Find the value of qr(2 - p)

To find the value of qr(2-p), notice that the expression can be simplified as follows: qr(2-p) = $$2 - \frac{\log{a}}{\log{(2a)}} = \frac{2\log{(2a)} - \log{a}}{\log{(2a)}}$$ Now using the properties of logarithms, \(\log{(2a)} = \log{2} + \log{a}\) Substituting this back into the expression, we get: qr(2-p) = $$\frac{2(\log{2} + \log{a}) - \log{a}}{\log{2} + \log{a}}$$ Simplifying further, we get:$$qr(2-p) = \frac{2\log{2} + \log{a}}{\log{2} + \log{a}}$$ Cancelling out the common term in the numerator and denominator, qr(2-p) = 2 Thus, the correct answer is (3) 2.

Key concepts

Change of Base Formula

The change of base formula is a fundamental aspect of logarithms that allows us to convert a log of one base into a log of another base. This is particularly useful when dealing with calculators that only compute logarithms for specific bases, such as base 10 (common logarithms) or base e (natural logarithms).

The formula is expressed as:\[\begin{equation}\log_{b}a = \frac{\log_{k}a}{\log_{k}b},\end{equation}\]where \( \log_{b}a \) is the log of \( a \) with base \( b \), and \( k \) can be any positive value, but typically we choose 10 or e for convenience. This property helps simplify logarithmic expressions and makes certain calculations possible when otherwise they would be very difficult or even impossible.

Logarithmic Identities

Logarithmic identities, or properties, are rules that simplify and enable us to work with logarithmic expressions. Some of the essential identities are the product, quotient, and power rules.

• Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \), where the logarithm of a product is the sum of the logarithms.
• Quotient Rule: \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \), which indicates that the logarithm of a quotient is the difference of the logarithms.
• Power Rule: \( \log_b(m^n) = n\log_b(m) \), showing that the logarithm of a power is the exponent times the logarithm of the base.

Remember, to apply these identities correctly, the base \( b \) must remain the same for all terms involved in the identity. These identities are critical when simplifying complex logarithmic expressions or converting them into a form that is more easily managed or solved.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions is an important skill that entails using logarithmic identities to rewrite logs more compactly and in a form that is easier to interpret or solve. Simplification often involves steps like:

• Combining logs using the product, quotient or power rules,
• Expanding logarithmic expressions,
• Applying the change of base formula where necessary, and
• Cancelling terms that appear in both the numerator and the denominator of a fractional logarithmic form.

In the context of our exercise, to find the value of \( qr(2 - p) \), it was essential to express the variables in a form that allowed for simplification. This process utilized the change of base formula and then applied the product and quotient rules of logarithms. The final step involved recognizing and cancelling out terms that appeared both in the numerator and denominator, ultimately leading us to the simplified result of the expression.