Question

log x to the base a log x to the base b divided log x to the base A + log x to the base b​

Answer 1

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Answer 2

Concept

We will use the properties of the logarithm and then try to simplify the given equation which is as follows,

(log x with base a + log x with base b)/(log x with base a * log x with

base b)

The properties of the logarithm is given as,

log a with base c + log b with base c = log ab with base c

Also,

Log m with base n = 1/ (log n with base m)

Given

The given expression is as follows,

(log x with base a + log x with base b)/(log x with base a * log x with

base b)

Find

We have to simplify the above given expression.

Solution

Since,

(log x with base a + log x with base b)/(log x with base a * log x with

base b)

Therefore,

(log x with base a)/(log x with base a * log x with base b) + (log x with base b)/(log x with base a * log x with base b)

= 1/( log x with base b) +1/(log x with base a)

Using the property, log m with base n = 1/ (log n with base m)

= log b with base x + log a with base x

Now, using the property, log a with base c + log b with base c = log ab with base c

=log ab with base x.

Hence the simplified value of the given expression is log ab with base x.

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