What is the exponential form of ${log}_{b} 35 = 3$ $log_b 35=3$ ?

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Answer 1 · 2018-06-07T14:44:35

$b^{3} = 35$ $b^3=35$

Lets start with some variables

If we have a relation between $a, b, c$ $a," "b," "c$ such that
$a = b^{c}$ $color (blue)(a=b^c$

If we apply log both sides we get

$log a = {log b}^{c}$ $loga=logb^c$

Which turns out to be

$log a = c log b$ $color (purple)(loga=clogb$

Npw divding both sides by $log b$ $color (red)(logb$

We get

$color (green)(loga/logb=c* cancel(logb)/cancel(logb)$

[Note: if logb=0 (b=1) it would be incorrect to divide both sides by $logb$ ... so $log_1 alpha$ isn't defined for $alpha!=1$ ]

Which gives us $color (grey)(log_b a=c$

Now comparing this general equation with the one given to us...
$color (indigo)(c=3$
$color (indigo)(a=35$

And so, we again get it in form
$a=b^c$

Here
$color (brown)(b^3=35$

What is the exponential form of log_b 35=3logb35=3log_b 35=3?