In this logarithmic problem, a logarithmic expression is given and it contains two terms but they are connected by a minus symbol. Each term in this logarithmic expression is in rational form.

$\dfrac{\log_{4}{17}}{\log_{9}{23}}$ $-$ $\dfrac{\log_{2}{17}}{\log_{3}{23}}$

Compare the expressions in the numerators of the two terms and also compare the expressions in the denominators of the two terms. It givens us an idea for simplifying the given logarithmic expression.

The $\log_{4}{(17)}$ and $\log_{2}{(17)}$ are the expressions in the numerators of the terms. They are in same form and their bases are different but $\log_{4}{(17)}$ can be transformed into $\log_{2}{(17)}$ because the base of expression $\log_{4}{(17)}$ can be expressed in the form of base of expression $\log_{2}{(17)}$. Similarly, the base of $\log_{9}{(23)}$ can also be expressed in the form of the base of $\log_{3}{(23)}$ by the exponentiation.

$=\,\,\,$ $\dfrac{\log_{2^2}{17}}{\log_{3^2}{23}}$ $-$ $\dfrac{\log_{2}{17}}{\log_{3}{23}}$

In the first term of the logarithmic expression, the bases of the logarithmic expressions in the numerator and denominator are in exponential notation. Hence, the exponent at base position of each logarithmic expression in the first logarithmic term can be separated by the base power rule of logarithms.

$=\,\,\,$ $\dfrac{\dfrac{1}{2} \times \log_{2}{17}}{\dfrac{1}{2} \times \log_{3}{23}}$ $-$ $\dfrac{\log_{2}{17}}{\log_{3}{23}}$

Now, we have to simplify the logarithmic expression for evaluating its value mathematically.

$=\,\,\, \require{cancel}$ $\dfrac{\cancel{\dfrac{1}{2}} \times \log_{2}{17}}{\cancel{\dfrac{1}{2}} \times \log_{3}{23}}$ $-$ $\dfrac{\log_{2}{17}}{\log_{3}{23}}$

$=\,\,\,$ $\dfrac{1 \times \log_{2}{17}}{1 \times \log_{3}{23}}$ $-$ $\dfrac{\log_{2}{17}}{\log_{3}{23}}$

$=\,\,\,$ $\dfrac{\log_{2}{17}}{\log_{3}{23}}$ $-$ $\dfrac{\log_{2}{17}}{\log_{3}{23}}$

The two terms in the above logarithmic expression are same. So, the value of first term should be equal to the value of second term. Hence, they both get cancelled each other mathematically.

$=\,\,\,$ $\cancel{\dfrac{\log_{2}{17}}{\log_{3}{23}}}$ $-$ $\cancel{\dfrac{\log_{2}{17}}{\log_{3}{23}}}$

$=\,\,\, 0$

Latest Math Topics

Jul 24, 2022

Jul 15, 2022

Latest Math Problems

Jul 29, 2022

Jul 17, 2022

Jun 02, 2022

Apr 06, 2022

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved