Solveeit Logo
Login

Question

Question: Which is greater \( x={{\log }_{3}}5\text{ or y=lo}{{\text{g}}_{17}}25\) ....

Which is greater x=log⁡35 or y=log1725 x={{\log }_{3}}5\text{ or y=lo}{{\text{g}}_{17}}25 .

Explanation

Solution

Hint: Start with the simplification of y using the identity log⁡bac=clog⁡ba=log⁡b1ca {{\log }_{b}}{{a}^{c}}=c{{\log }_{b}}a={{\log }_{{{b}^{\dfrac{1}{c}}}}}a . Once you are done with the simplification, to compare between logarithmic functions, you need to ensure that they have the same base. So, use the identity log⁡ba=1log⁡ab {{\log }_{b}}a=\dfrac{1}{{{\log }_{a}}b} to get the logarithmic functions with the same base. Once you get two logarithmic functions with the same base, the function with higher value of x in the form log x is greater, provided base is greater than 1.

Complete step-by-step answer:
Let us start the simplification of y using the identity log⁡bac=clog⁡ba=log⁡b1ca {{\log }_{b}}{{a}^{c}}=c{{\log }_{b}}a={{\log }_{{{b}^{\dfrac{1}{c}}}}}a .
y=log⁡1725=log⁡1752=2log⁡175=log⁡17125y={{\log }_{17}}25={{\log }_{17}}{{5}^{2}}=2{{\log }_{17}}5={{\log }_{{{17}^{\dfrac{1}{2}}}}}5
Now, we will use the identity log⁡ba=1log⁡ab {{\log }_{b}}a=\dfrac{1}{{{\log }_{a}}b} for both x and y to make the bases of the logarithmic part same, i.e., 5. On doing so, we get
y=log⁡17125=1log⁡51712y={{\log }_{{{17}^{\dfrac{1}{2}}}}}5=\dfrac{1}{{{\log }_{5}}{{17}^{\dfrac{1}{2}}}}
x=log⁡35=1log⁡53x={{\log }_{3}}5=\dfrac{1}{{{\log }_{5}}3}
Now, if we compare log⁡51712 {{\log }_{5}}{{17}^{\dfrac{1}{2}}} and log⁡53 {{\log }_{5}}3 , we know , the function with higher value of x in the form log x is greater, provided base is greater than 1 and bases are same. So, 1712 {{17}^{\dfrac{1}{2}}} is for sure greater than 4 as the square of 4 is 16 which is less than 17. So, log⁡51712 {{\log }_{5}}{{17}^{\dfrac{1}{2}}} is greater than log⁡53 {{\log }_{5}}3 .
Now, as the two terms are the denominators of x and y, we can say that denominator of y is greater than x and both x and y have the same numerator. So, we can conclude that x>y.
Therefore, the answer to the above question is x>y x>y .

Note: The most important thing to remember is that if two fractions have the same numerator and the denominator is different, then the fraction with the smaller denominator is greater, but the condition attached to this is both numerator and denominator needs to be positive and as the value of logarithmic function of a number is positive if the number and the base is greater than 1 which is true in the above solution.