If logxa, ax and logbx are in GP, then what is x equal to ?
The problem states that the terms log$_x$a, ax, and log$_b$x are in Geometric Progression (GP). In a GP, the square of the middle term is equal to the product of the first and third terms.
However, the second term is given as 'ax'. Based on the structure of the options and the likely intended mathematical problem, it is highly probable that the second term was intended to be \(a^x\) instead of the product \(a \times x\). We will proceed with the assumption that the terms in GP are \(\text{log}_x\text{a}\), \(a^x\), and \(\text{log}_b\text{x}\).
Understanding Geometric Progression and Logarithms
A sequence of non-zero numbers is in Geometric Progression if the ratio of any term to its preceding term is constant. This constant ratio is called the common ratio. If three terms \(t_1, t_2, t_3\) are in GP, then \(t_2/t_1 = t_3/t_2\), which implies \(t_2^2 = t_1 \cdot t_3\). The term \(t_2\) is the geometric mean of \(t_1\) and \(t_3\).
We will also use fundamental logarithm properties, especially the change of base formula: \(\text{log}_p q = \frac{\text{log}_r q}{\text{log}_r p}\) for any valid base \(r\). A common base like the natural logarithm (\(\ln\)) or base 10 logarithm (\(\text{log}_{10}\)) is often used. We will also use \(\text{log}_p p^q = q\) and \(p^{\text{log}_p q} = q\).
Setting up the GP Equation
Assuming the terms are \(\text{log}_x\text{a}\), \(a^x\), and \(\text{log}_b\text{x}\) in GP, the condition is:
\[ (a^x)^2 = (\text{log}_x\text{a}) \cdot (\text{log}_b\text{x}) \]Solving for x using Logarithm Properties
Let's simplify both sides of the equation.
Left Hand Side (LHS):
\[ (a^x)^2 = a^{2x} \]Right Hand Side (RHS):
Using the change of base formula (e.g., to natural logarithm \(\ln\)):
\[ \text{log}_x\text{a} = \frac{\ln a}{\ln x} \] \[ \text{log}_b\text{x} = \frac{\ln x}{\ln b} \]So, the product is:
\[ (\text{log}_x\text{a}) \cdot (\text{log}_b\text{x}) = \left(\frac{\ln a}{\ln x}\right) \cdot \left(\frac{\ln x}{\ln b}\right) \]Assuming \(\ln x \ne 0\) (which implies \(x \ne 1\)), we can cancel \(\ln x\):
\[ \left(\frac{\ln a}{\ln x}\right) \cdot \left(\frac{\ln x}{\ln b}\right) = \frac{\ln a}{\ln b} \]Using the change of base formula in reverse, \(\frac{\ln a}{\ln b} = \text{log}_b\text{a}\). So, the RHS simplifies to \(\text{log}_b\text{a}\).
Equating the simplified LHS and RHS:
\[ a^{2x} = \text{log}_b\text{a} \]To solve for \(x\), take the logarithm base \(a\) on both sides of the equation:
\[ \text{log}_a(a^{2x}) = \text{log}_a(\text{log}_b\text{a}) \]Using the logarithm property \(\text{log}_p p^q = q\) on the left side:
\[ 2x = \text{log}_a(\text{log}_b\text{a}) \]Finally, divide by 2 to find the value of \(x\):
\[ x = \frac{\text{log}_a(\text{log}_b\text{a})}{2} \]Final Value of x
Based on the assumption that the second term in the GP is \(a^x\), the value of \(x\) is \(\frac{\log _a\left(\log _b a\right)}{2}\).
Revision Table - Logarithm and GP Concepts
| Concept | Property / Definition |
|---|---|
| Geometric Progression (GP) | If \(t_1, t_2, t_3\) are in GP, then \(t_2^2 = t_1 \cdot t_3\). |
| Logarithm Change of Base | \(\text{log}_p q = \frac{\text{log}_r q}{\text{log}_r p}\) |
| Logarithm Power Rule | \(\text{log}_p p^q = q\) |
| Logarithm Property | \(p^{\text{log}_p q} = q\) |
Additional Information - Properties of Logarithms and GP
Geometric Progression (GP): A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). The general form is \(a, ar, ar^2, ar^3, \dots\). If three terms \(t_1, t_2, t_3\) are consecutive terms in a GP, then the ratio \(t_2/t_1 = t_3/t_2 = r\), leading to \(t_2^2 = t_1 \cdot t_3\).
Logarithm Properties:
- Product Rule: \(\text{log}_b (MN) = \text{log}_b M + \text{log}_b N\)
- Quotient Rule: \(\text{log}_b (M/N) = \text{log}_b M - \text{log}_b N\)
- Power Rule: \(\text{log}_b (M^p) = p \cdot \text{log}_b M\)
- Change of Base Formula: \(\text{log}_b a = \frac{\text{log}_c a}{\text{log}_c b}\)
- Inverse Property: \(\text{log}_b b = 1\) and \(\text{log}_b 1 = 0\)
- Relationship with Exponents: \(\text{log}_b a = c \iff b^c = a\)
These properties are crucial for simplifying logarithmic expressions and solving equations involving logarithms, as demonstrated in this problem.
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