Question

If $x,y,z$ are in G.P. and $a^x=b^y=c^z$, then : (a) ${\log }_{b}a={\log }_{c}b$ (b) ${\log }_{b}a={\log }_{a}c$ (c) ${\log }_{c}b={\log }_{a}c$ (d) none of these

Short answer

(a) ${\log }_{b}a={\log }_{c}b$

Step-by-step solution

Express x, y, and z in terms of a common ratio

Since x, y, and z are in G.P., let the common ratio be 'r'. Then, we can write: - y = x * r - z = y * r = x * r^2

Apply logarithms and logarithmic properties

Now, let's use the equation $a^x=b^y=c^z$ and apply logarithms to simplify it. Since all of the expressions are equal, we can use one as a base and apply logarithms to the others. Let's use $a^x$ as a base: - ${\log }_{a^x}{a}^x={\log }_{a^x}{b}^y={\log }_{a^x}{c}^z$ Using the logarithmic properties, we can simplify each term: - $x{\log }_{a}a=y{\log }_{a^x}b=z{\log }_{a^x}c$ Since ${\log }_{a}a=1$: - $x=y{\log }_{a^x}b=z{\log }_{a^x}c$ Now, let us replace y and z using the common ratio from Step 1: - $x=xr{\log }_{a^x}b=x{r}^2{\log }_{a^x}c$ Cancel 'x' from all terms: - $1=r{\log }_{a^x}b=r^2{\log }_{a^x}c$

Analyze the relationships between logarithms

From the equation above, we have: - $r{\log }_{a^x}b=1$ - $r^2{\log }_{a^x}c=1$ Notice that if we take the first equation and multiply it with ${\log }_{c}a$ we get: - $r{\log }_{a^x}b\cdot {\log }_{c}a={\log }_{c}a$ - $r{\log }_{a^x}={\log }_{c}ba$ (using the log properties) Now, we can find the relationship between the logarithms by comparing the equations derived in this step. Between equations: - $r{\log }_{a^x}b=1$ and $r{\log }_{a^x}={\log }_{c}ba$ we can conclude: ${\log }_{b}a={\log }_{c}b$ Therefore, the correct answer is (a).

Key concepts

Logarithmic Properties

Logarithmic properties are essential tools in simplifying complex equations and solving mathematical problems. When you're dealing with equations like $a^x=b^y=c^z$, applying logarithms can make the expressions easier to manage.

Here's how it works:

• The power rule of logarithms states that ${\log }_b(m^n)=n\cdot {\log }_{b}m$. This is useful for bringing exponents, like $x$, $y$, or $z$, to the front of the logarithm.
• The base change formula, ${\log }_{b}m=\frac{{\log }_{k}m}{{\log }_{k}b}$, allows you to convert logarithms between different bases.
• The identity ${\log }_{b}b=1$ often simplifies calculations as we see when ${\log }_{a}a=1$.

By applying these properties, we can simplify and compare expressions to identify potential solutions, such as concluding that ${\log }_{b}a={\log }_{c}b$.

Exponential Equations

Exponential equations feature variables in the exponent and can appear daunting at first glance. However, they can be elegantly tackled using logarithmic transformations. In the example given, $a^x=b^y=c^z$ is an exponential equation where the expressions are equal and can be equated to a single variable, making the system consistent.

• Firstly, recognize that all parts are equal, allowing us to express each component in terms of the other. For example, $a^x$ can serve as a reference to express $b^y$ and $c^z$.
• Applying logarithms to both sides helps remove the variables from the exponents, transforming the exponential forms into linear forms, which are simpler to solve.
• This transformation enables the use of equivalent relationships to find consistent solutions, such as ${\log }_{b}a={\log }_{c}b$ by analyzing the transformed logarithmic expressions.

Exponential equations thus transform from the complexity of exponents to the manageability of linear equations, making solutions much easier to find.

Mathematical Reasoning

Mathematical reasoning is a systematic approach to solving problems and arriving at logical conclusions. It often involves identifying patterns, applying known mathematical properties, and making deductions based on established relationships. In this exercise, mathematical reasoning is used in the following way:

• First, recognizing that $x,y,z$ form a geometric progression provides information about their multiplicative relationships, allowing us to express them in terms of a common ratio $r$.
• Reasoning through the application of logarithms simplifies the equation, allowing us to make logical connections between base and result, leading to the equation ${\log }_{b}a={\log }_{c}b$.
• Throughout, substituting variables from step to step based on earlier deductions ensures coherence and validity of the final conclusions, which speak to the power of mathematical reasoning in providing clear solutions.

Mathematical reasoning not only helps in understanding individual problems but also in forming strategies for tackling a wide range of mathematical challenges.