What's the point of using logarithms when sketching the inverse of a function?

[maki1995]

What's the point of logarithms when trying to sketch functions? Isn't y=3^x the same as x=3^y? I think it should be the same, but I get different results for each method. If it is the same, what's the point of y=log3x? It's confusing and I don't get the same results when trying to find the points in the graph. It makes sense when you try to solve an equation with different bases, but what's the point of logarithms when sketching graphs?
For example, let's say you have y=3^x. The inverse function would be y=log3x or x=3^y. I'm trying to find the points for the graph of the inverse function. Let's say y=2. 2=log3x which means x would be 4.19. Now, for the other equation, if x=3^2, x would be 9, which doesn't make sense.

 

[AlephZero]

Isn't y=3^x the same as x=3^y?

No it's not. The inverse function of an exponential function is a log function. That's the "point" of logs.

Saying y=3^x the same as x=3^y as just as wrong as saying y = x^3 is the same as x = y^3.

 

[Mute]

When you plot a function y =vs. x, there is a distinction between y and x: x is typically the "independent variable" and y is the "dependent variable". They are called this because we pick x to be whatever we want, but there is some relation y = f(x) that determines y - hence y depends on x, but x is independent of y. Now, the labels are aribitrary - you could be y to be the independent variable and x to be the dependent variable, as you will see in a moment.

When you write "y = 3^x", x is independent and y is dependent. So, we can pick x and find out y. But, suppose we wanted to pick y, and figure out what x needs to be to get that particular y we want. What that means is we want to find the function x = g(y), such that 3^x = 3^g(y) = y, where y is now the independent variable and x is the dependent variable. However, the labels are arbitrary, and people usually like x to be dependent and y to be independent, so we switch the labels x and y to get 3^y = 3^g(x) = x.

The result is that when you write y = 3^x, x is independent and y is dependent. When you are talking about finding the inverse relation to the function and you write x = 3^y, x is still independent and y is dependent, so in the two cases y actually represents two different functions. In the first case, y represents the function y = f(x) = 3^x. In the second case, y represents the function y = g(x) = log_3(x).

Does that make things clearer, or should I try to explain it again?

 

[Mark44]

The point of inverse functions is that if you have y as some function of x (i.e., y = f(x)), the inverse allows you to write x as a different function of y.

For example, if y = 10^x, if you're given a value of x, you can compute the associated value of y fairly easily. However, if you're given a value of y, it's not so easy to find the associated x value, unless you know that the inverse of the 10^x function is the log₁₀ function.

More precisely, the equation y = 10^x is equivalent to x = log₁₀(y). These two equations have exactly the same graph, meaning that any ordered pair (x, y) that satisfies one equation, also satisfies the other.

 

[symbolipoint]

Another explanation similar to what Mark44 said, if you have a set of two-dimensional data points which may fit y=10^x, and you know this is a function, then in case its inverse is also a function (which certainly in that form, it is), then you can use x as a function of y, and show x=log₁₀(y). You might choose this if you take y as the independent variable and x as the DEPENDENT variable. In fact, you might choose variable names other than "x" and "y". Maybe pairs, (s, t), or (q, r), or (n, p), ... whatever.