A passage in my textbook has me confused, first it states this:
$ \log _{a}\left( x\right) .\log _{b}\left( a\right) =\log _{b}\left( a^{\log _{a}\left( x\right) }\right) =\log _{b}\left( x\right) $
I understand this to be a form of the change of base rule.
But it then goes onto say:
"To change the base of an exponential expression, we multiply the input by the factor $log _{b}\left( a\right)$. To change the base of a logarithmic expression, we divide the output by the factor $log _{b}\left( a\right)$. ...What this tells us is that all exponential and logarithmic functions are scalings of one another."
My questions are:
(1) What does the text mean by: "to change the base of an exponential expression you need to multiply the input by the factor $log _{b}\left( a\right)$"?
I know it refers to the formula: $log _{a}\left( x\right) .\log _{b}\left( a\right) =\log _{b}\left( x\right)$, but what does it mean by "to change the base of an exponential expression"? Isn't this formula meant so we can change the base b of a "logarithmic expression"? This is confusing.
(2) What does the text mean when it says: "this tells us that all exponential and logarithmic functions are scalings of one another"?
I'd love for this expanded upon, it leaves a lot to be desired.
Thank you.
(1) An exponential expression is something like $a^x$. In going from $a^x$ to $b^{\log_b(a) x}$ you are multiplying the input $x$ by $\log_b(a)$.
(2) $\log_b(x) = \log_b(a) \log_a(x)$ says that the graph of $y = \log_b(x)$ is obtained from the graph of $y = \log_a(x)$ by scaling the $y$ axis, i.e. multiplying $y$ by a positive (assuming $a, b > 1$) constant.
$a^x = b^{\log_b(a) x}$ says that the graph of $y = a^x$ is obtained from the graph of $b^x$ by scaling the $x$ axis, i.e. multiplying $x$ by a positive (again assuming $a,b > 1$) constant.