hmm
47^x = 30
log_47 (47^x) = log_47 (30)
x = log_47 (30)
A concept to consider here is the *inverse* of a function.
What is an inverse of a function?
In a nutshell, if two functions f and g are inverses of each other, then if we call f with the output of g evaluated for some input, we should get that input back in return; and likewise, if we call g with the output of f evaluated for some input, we should get that input back in return.
So simple example, say f(x) and g(x) are inverses, then what is f(g(x)) and g(f(x))?
They’re both x.
With this, a simpler notation for the above you might see is, (f o g)(x) = f(g(x)) = x iff f and g are inverses. Likewise, (g o f)(x) = g(f(x)) = x iff f and g are inverses.
So now then, your situation
47^x = 30
log_47 (47^x) = log_47 (30)
x = log_47 (30)
How did we know to do the 2nd step?
Well it turns out, an exponential function and log function are inverses if the base of the exponential is the same base as the log.
For example, we have 47^x = 30
The base of this exponential is 47. While there is no log, we can make one, where it’s base is also 47 and do this to both sides!
log_47 (47^x) = log_47 (30)
Now, recall the exponential and log now are inverses, they have the same base. We could then think of this in terms of the other inverses we talked about.
Say f(x) = 47^x and g(x) = log_47 (x)
Then, f(g(x)) = g(f(x)) = x.
That is, the rule you see above is taking the log of each side with the same base of the exponent, that is g(f(x)) = log_47 (47^x) = x
Which is why you saw the final step
log_47 (47^x) = log_47 (30)
x = log_47 (30)
The other rule, not seen in this problem is this one, f(g(x)) = 47^(log_47 (x)) = x. This is called the **exponentiation** rule.
If we exponentiate - make a base that is the same as that of the log’s base and raise that base to the power of the log, then we get x (whatever is inside the parenthesis).
So hopefully that puts more clarification on the table. What is going on, how did we know to do that rule, where that rule comes from / relates to.
