Question

If \(\mathrm{p}, \mathrm{q}, \mathrm{r}\) are in geometric progression and \(\mathrm{a}^{\mathrm{p}}=\mathrm{b}^{\mathrm{q}}=c^{t}\), then which of the following is true? (1) \(\log _{c} b_{-} \log _{a} c\) (2) \(\log _{c} \mathrm{~b}_{-} \log _{\mathrm{b}} \mathrm{a}\) (3) \(\log _{c} a_{-} \log _{b} c\) (4) None of these

Short answer

(1) \(\log_c{b} - \log_a{c} = \frac{q}{p} - \frac{p}{r}\) (2) \(\log_c{b} - \log_b{a} = \frac{q}{p} - \frac{1}{\log_a{b}}\) (3) \(\log_c{a} - \log_b{c} = \frac{p}{r} - \frac{q}{p}\) (4) None of these Answer: (4) None of these

Step-by-step solution

Write the given information

We know the variables p, q, and r are in geometric progression, and we are given the equation: \[a^p = b^q = c^r\]

Find the common ratio

Since p, q, and r are in geometric progression, we know there is a common ratio. So we have: \[q = pr\] With p, q, r > 0.

Convert the exponential equations to logarithm equations

Using the property of logarithms, we can rewrite the given equation as follows: \[\log_a{a^p} = \log_a{b^q} = \log_a{c^r}\] Now applying the power rule of logarithms, we simplify the equations: \[p\log_a{a} = q\log_a{b} = r\log_a{c}\] Since \(\log_a{a} = 1\), the equations become: \[p = q\log_a{b} = r\log_a{c}\]

Organize the logarithmic equations

From Step 3, we have two equations: \[\log_a{b} = \frac{p}{q}\] \[\log_a{c} = \frac{p}{r}\] Let's organize the given options using the relationship \(\log_a{b} = \frac{p}{q}\) and \(\log_a{c} = \frac{p}{r}\): (1) \(\log_c{b} - \log_a{c} = \frac{q}{p} - \frac{p}{r}\) (2) \(\log_c{b} - \log_b{a} = \frac{q}{p} - \frac{1}{\log_a{b}}\) (3) \(\log_c{a} - \log_b{c} = \frac{p}{r} - \frac{q}{p}\)

Evaluate the options

Now, we know that \(q=pr\), substituting \(q\) in equations (1) and (3): (1) \(\frac{pr}{p} - \frac{p}{r} = r - \frac{p}{r}\) (3) \(\frac{p}{r} - \frac{pr}{p} = \frac{p}{r} - r\) Now we can clearly see that: (1) is equal to (3), so there is no unique relationship. (2) does not match any of the equations from Step 4. Thus, the answer is: (4) None of these

Key concepts

Logarithmic Equations

Understanding logarithmic equations is essential when dealing with exponential expressions in a geometric progression. To decipher a logarithmic equation, one must remember that it represents the exponent to which a base must be raised to produce a given number. For example, if we have the equation \( b^q = a^p \), converting this to a logarithmic equation yields \( q\log_b{a} = p \), displaying that exponent \( q \) is what we need to raise \( b \) to get \( a^p \).

In the given exercise, we utilized this very concept to manipulate the given exponential equations into a solvable logarithmic equation, where \( p \), \( q \) and \( r \) are in a geometric progression, and \( a^p = b^q = c^r \). By methodically dissecting this relationship through logarithms, we transformed the exponent problem into an equation we could easily handle with our knowledge of logarithmic properties.

Exponents to Logarithms

Converting exponents to logarithms streamlines solving equations where the unknown is the exponent itself. The fundamental base of this transformation is \( b^x = y \), where \( x \) can be isolated by rewriting it as \( x = \log_b{y} \). By employing this logic, we are taking advantage of logarithms to resolve expressions that would be otherwise difficult to manage.

In our exercise, we witnessed this by translating \( a^p = b^q = c^r \) into logarithmic form. This conversion facilitates the use of the properties of logarithms to solve for the variables. As the exercise shows, understanding how to fluctuate between exponents and logarithms is paramount in simplifying complex algebraic equations and finding a clear solution.

Properties of Logarithms

Logarithms possess distinct properties that make them exceedingly useful in algebra. The most commonly used properties include the product rule, the quotient rule, and the power rule. The product rule states that \( \log_b(xy) = \log_b(x) + \log_b(y) \), and the quotient rule says \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \). The power rule, which we employed in our exercise, informs us that \( \log_b(x^y) = y\log_b(x) \), a property that is pivotal in reducing complex expressions.

Applying these principles, we managed to condense our initial equation into a more workable form. This allowed us to explore different logs such as \( \log_a{b} \) and \( \log_a{c} \) against the constraints of the geometric progression. Harnessing the properties of logarithms, we unraveled the given equations and logically ruled out the incorrect answer choices.