Root in Base, Power in Base

Algebra Level 2

Is $\color{#69047E}{ \log_{b^{1/n}} x^m}$ and $\color{#3D99F6}{\log_{b^m} x^{1/n}}$ equivalent?

Originally by Sir @Martin Soliman

Try this one

Yes Not enough information Flawed Question No

3 solutions

Paul Ryan Longhas
Mar 24, 2015

Since, $log_{b^ {\frac{1}{n}}}x^m$ and $log_{b^m}x^{ \frac{1}{n}}$

$=> \frac{logx^m}{logb^{1/n}} =?= \frac{logx^{1/n}}{logb^m}$ $=> \frac{mlogx}{1/n(logb)} =?= \frac{1/n(logx)}{mlogb}$ $=> \frac{m}{\frac{1}{n}} =?= \frac{ \frac{1}{n}}{m}$ $=> mn \neq \frac{1}{mn}$ Therefore,, $No$ .

It may hold true for m= '1/n' ,and m=n=1 ; so why can't we opt "Incomplete Information" ?

Garima Garima - 6 years, 2 months ago

They ask whether the expressions are "equivalent", meaning that the value is the same for ALL choices of the variables in the domain.

Otto Bretscher - 6 years, 2 months ago

But I suppose variable here should be 'x' sir, and not 'm' or 'n'

Garima Garima - 6 years, 2 months ago

@Garima Garima – It is all a matter of definitions and conventions, of course. In the customary terminology, an (algebraic) expression may contain numbers, variables, operations, and exponents... so, sure, $x, m, n$ and $b$ are all variables. Why would you say that $x$ is a variable but $m$ isn't?

Otto Bretscher - 6 years, 2 months ago

@Otto Bretscher – whoops . Perhaps....

Garima Garima - 6 years, 2 months ago

@Garima Garima- I thought the same thing and got wrong

Natalia Dcruz - 3 years, 3 months ago

But if m and n are 1, then both log values are equal. So sometimes should be the answer.

Suresh Patel - 6 years, 2 months ago

It is very rare (if ever) that "sometimes correct" is an answer for a math question. More often than not (if not always) it's a true or false situation. Had the question asked if the expressions were equivalent when m=n=1, then it would be true. But since the question didn't specify the values of m or n, your answer also cannot specify that.

Louis W - 6 years, 2 months ago

If mn = 1, then it is true. I think 'not enough information' should be the answer.

Bhargav Upadhyay - 6 years, 2 months ago

Otto Bretscher
Mar 25, 2015

We give a counterexample: $b=x=10, n=1, m=2$ . Now $\log_{b^{1/n}} x^m= \log_{10} (100)=2$ and $\log_{b^m} x^{1/n}=\log_{100} 10=\frac{1}{2}$ .

Samarth Agarwal
Mar 28, 2015

Anything of form : $log_a^x b^y$ can be written as: $\frac {y}{x}$ $\log_a b$ . Therefore, $\log_b^\frac {1}{n} x^m$ can be written as: $m \times n$ $\log_b x$ and $log_b^m x^\frac {1}{n}$ can be written as: $\frac {1}{m \times n}$ $\log_b x$ . Thus they are not same.(Please ignore errors as I am new to LaTeX code).

Root in Base, Power in Base

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