True or false? Justify your answer.
b^(log of x to the base b) = x
I don't even know solve for this myself. Could someone please show their work for this?
Given that $\displaystyle b,c \neq 0,1 $ and $\displaystyle x > 0$ so the domain is satisfied
Suppose the following is true: $\displaystyle b^{log_b(x)} = x$
Take the log of both sides
$\displaystyle log_b(x)log_c(b) = log_c(x)$
From the change of base rule $\displaystyle log_b(x) = \frac{log_c(x)}{log_c(b)}$
$\displaystyle \frac{log_c(x)}{log_c(b)} \, log_c(b) = log_c(x)$
$\displaystyle log_c(b)$ cancels to give $\displaystyle log_c(x)=log_c(x)$
Therefore it is true.
I'm curious- what is your definition of "$\displaystyle log_b(x)$"?
(The reason I'm curious is that what you stated is generally given as the definition! If you are using that definition, "it is true by the definition of $\displaystyle log_b(x)$" would be a perfectly good answer!)