Math Doubts

Find the value of $\log_{d} abc$ if $a^2 \,=\, b^3 \,=\, c^5 \,=\, d^6$

$a$, $b$, $c$ and $d$ are four literals. The square of $a$, cube of $b$, fifth power of $c$ and sixth power of $d$ are equal.

$a^2 \,=\, b^3 \,=\, c^5 \,=\, d^6$, then the value of logarithm of product of $a$, $b$ and $c$ to base $d$ is required to find and it is written in logarithms as $\displaystyle \log_{d} abc$ mathematically.

01

Power to Root Transformation

The logarithm of product of literals $a$, $b$ and $c$ is evaluated on the basis of literal number $d$. Hence, transform the exponents of literal numbers $a$, $b$ and $c$ in terms of $d$ purely.

$a^2 \,=\, d^6 \,\implies\, a \,=\, {(d^6)}^{\dfrac{1}{2}}$

$b^3 \,=\, d^6 \,\implies\, b \,=\, {(d^6)}^{\dfrac{1}{3}}$

$c^5 \,=\, d^6 \,\implies\, c \,=\, {(d^6)}^{\dfrac{1}{5}}$

02

Substitution

Substitute the value of $a$, $b$ and $c$ in terms $d$ in the logarithmic term.

$\displaystyle \log_{d} abc$ $\,=\,$ $\displaystyle \log_{d} \Big[{(d^6)}^{\dfrac{1}{2}} {(d^6)}^{\dfrac{1}{3}} {(d^6)}^{\dfrac{1}{4}}\Big]$

03

Simplification

$d^6$ is an exponential term and it is a common base for the three exponential terms. So, use the product rule of exponents to simplify it.

$=\,\,\,$ $\log_{d} {(d^6)}^{\dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{5}}$

$=\,\,\,$ $\log_{d} {(d^6)}^{\dfrac{15+10+6}{30}}$

$=\,\,\,$ $\log_{d} {(d^6)}^{\dfrac{31}{30}}$

The exponential term also has another exponent. It can be simplified by the power rule of power of an exponential term.

$=\,\,\,$ $\log_{d} d^{\displaystyle 6 \times \dfrac{31}{30}}$

$=\,\,\,$ $\log_{d} d^{\dfrac{6 \times 31}{30}}$

$=\,\,\,$ $\require{cancel} \log_{d} d^{\dfrac{\cancel{6} \times 31}{\cancel{30}}}$

$=\,\,\,$ $\log_{d} d^{\dfrac{31}{5}}$

Use power rule of logarithms to simplify it further.

$=\,\,\,$ $\dfrac{31}{5} \times \log_{d} d$

According to logarithm of base rule, the value is one.

$=\,\,\,$ $\dfrac{31}{5} \times 1$

$=\,\,\,$ $\dfrac{31}{5}$



Follow us
Email subscription
Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Mobile App for Android users Math Doubts Android App
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more