# Simplify $a^\log_a} b . \log_b} c . \log_c} d . \log_d} x$

$a$, $b$, $c$, $d$ and $x$ are five literals. The product of the terms logarithm of $a$ to $b$, logarithm of $b$ to $c$, logarithm of $c$ to $d$ and logarithm $d$ to $x$ becomes an exponent to the literal number $a$.

$\large a^\log_a} b . \log_b} c . \log_c} d . \log_d} x$

The product of the logarithmic terms follows a sequence and it allows us to simplify the expression easily.

###### Step: 1

Consider the product of first two logarithmic terms.

$= \large a^(\log_a} b . \log_b} c) . \log_c} d . \log_d} x$

Now use change of base rule of logarithm to combine the product of them as a logarithmic term.

$= \large a^(\log_a} c ) . \log_c} d . \log_d} x$

$= \large a^\log_a} c . \log_c} d . \log_d} x$

###### Step: 2

Once again, consider the product of first two logarithm terms and use same change of base log rule to simply the product of them.

$= \large a^(\log_a} c . \log_c} d) . \log_d} x$

$= \large a^(\log_a} d) . \log_d} x$

$= \large a^\log_a} d . \log_d} x$

###### Step: 3

Similarly, repeat the same procedure one more time.

$= \large a^\log_a} d . \log_d} x$

$= \large a^\log_a} x$

###### Step: 4

There is only one logarithm term as an exponent of the literal a and the base of the logarithm is also same. It can be simplified by the fundamental logarithmic identity.

$= \large a^\log_a} x$

$= \large x$

Latest Math Topics
Latest Math Problems

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

###### Maths Topics

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

###### Maths Problems

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

Learn solutions

###### Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.