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I learnt that exponential functions has the format : \begin{align} number^x \end{align}

But these functions are not exactly power x, so I am confused:
\begin{align} \sqrt{5}^{\log_{5} x}\\ 8^{\lg x} \end{align}
I am trying to understand the classes of function, so I would like to know what class are the above 2 functions in

Mireodon
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2 Answers2

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Note that $$ \sqrt 5 ^{\log_5 x } =5^{1/2 \log_5 x}=\sqrt {x} $$

Which is not exponential.

Also $$8^{\log x} =x^{\log 8}$$

Which is not exponential

Thus none of your expressions is exponential.

Mohammad Riazi-Kermani
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  • Oh, d'oh! My answer is .... so dumb now! – fleablood Jul 14 '19 at 16:20
  • For what it's worth. In the OP it was $8^{\lg x}$ where $\lg = \log_2$ so $8^{\lg x} = x^{\lg 8} = x^3$. ..... In hindsight this was clearly from an exercise designed to give the student practice in converting bases... I can't believe it escaped me as badly as it did. – fleablood Jul 14 '19 at 16:36
  • I'm surprised neither of you really mentioned my favourite log law: $$a^{\log_c{(b)}}=b^{\log_c{(a)}}$$ – Peter Foreman Jul 14 '19 at 16:44
  • I used it in my answer without mentioning it. – Mohammad Riazi-Kermani Jul 14 '19 at 16:49
  • I always forget that law. Not sure why. If I remembered it more often it would have been obvious. My favorite law is $\log_c a = \frac 1{\log_a c}$... Anyway, ... two minutes before I read your comment I redid my answer entirely in terms of that law. – fleablood Jul 14 '19 at 17:42
  • Tangent: I think the reason I always forget that law is because expressions involving logs and exponents are questions of "blocking" and I usually just block to a standard base. $a^{\log_c b} = e^{\ln a\frac{\ln b}{\ln c}}$. There is usually little incentive to go back to $=e^{\ln b\frac {\ln a}{\ln c}} = b^{\log_c a}$.... I guess.... but I digress. – fleablood Jul 14 '19 at 17:48
  • It was mentioned to me by a student for the first time and I did not believe it. It is very beautiful. I proved it by taking logarithm of both sides. – Mohammad Riazi-Kermani Jul 14 '19 at 17:50
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In hindsight I think the text where you found these functions wanted you to use the identity:

$a^{\log_c b} = b^{\log_c a}$

(Proof: $(a)^{\log_c b} =$ $(c^{\log_c a})^{\log_c b}=$ $c^{\log_c a\cdot \log_c b}=$ $(c^{\log_c b})^{\log_c a} =$ $(b)^{\log_c a}$.)

So $\sqrt 5^{\log_5 x} = x^{\log_5 \sqrt 5} = x^{\frac 12} = \sqrt x$.

And $8^{\lg 2 x}= x^{\lg 8} = x^3$.

So ....

A function $b^{x}$ is an exponential function. A function $x^b$ is a power function. (They are different.) A function $b^{\log_k x}=x^{\log_k b}$ is a power function in disguise.

====== old answer =====

Okay....

An exponential function is of the form $b^x$. You are describing functions of the form $b^{f(x)}$ which are not exponential functions but clearly related.

In particular you are talking of functions of the form $b^{\log_k x}$.

Now we can change the base of an exponential function by noting that

$b^x = (b)^x = (k^{\log_k b})^x = k^{(x\cdot \log_k b}=k^{dx}$ where $d = \log_k b$. Or by noting $b^x = b^{ax*\frac 1a} = c^{ax} $ where $c = b^{\frac 1a}$.

So exponential functions can be in the form $b^{ax}$ for a constant $a$.

Your functions $f(x) = b^{\log_k x}$ can have a change of base as well.

$(b)^{\log_k x} = (k^{\log_k b})^{\log_k x} = k^{\log_k b\cdot \log_k x} = (k^{\log_k x})^{\log_k b} = x^{\log_k b} = x^c$ where $c = \log_k b$ is some constant.

So ....

Exponential functions are of the form $b^x$. Power functions are of the form $x^c$. Yours are actually examples of power functions.

[My first version overlooked the obvious and made a very mistaken conclusion. You can view the edit history to see a naive, not incorrect but utterly naive, earlier take about your functions.]

fleablood
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