If often hear log being referred to as the inverse function of exponentials. I also hear log base a of n is equivalent to a^x=n. I don't understand how it could be both, could someone explain?

Original post by quickquestion805

If often hear log being referred to as the inverse function of exponentials. I also hear log base a of n is equivalent to a^x=n. I don't understand how it could be both, could someone explain?

If you are defining x as log{a}(n) (intended to be read as log base a of n), then yes, a^x = n is correct.

Starting from your second statement:

a^x = n

Taking logarithms of base a of both sides:

log{a}(a^x) = log{a}(n)

Since log{a}(…) and a^(…) are defined as inverse functions, log{a}(a^x) must be the input of a^x, which is x (this is applying the rule that f(f’(x)) = x. This leads to:

x = log{a}(n)

I think it’s better to say that log{a}(n) = x <=> n = a^x (i.e the two statements imply one another) than the two statements are equivalent.

(edited 1 year ago)

Hmm... Not sure what your question is, but they are not contradictory. I'll try to spit out gibberish, in the hopes you have some sort of idea.

In a way it depends on what starting definition of log are you using, because one implies the other.

Say you start with "k=log_10(x) is the solution in 10^k=x" as your starting point, i.e. your second description of log.

What does inverse function mean? It means we are trying to find some function, let's call it L(x), such that 10^{L(x)} = x. By our starting definition, this L(x) could only be log_10(x) (since inverses are unique).

The other way, well... I'm lazy...

In a way it depends on what starting definition of log are you using, because one implies the other.

Say you start with "k=log_10(x) is the solution in 10^k=x" as your starting point, i.e. your second description of log.

What does inverse function mean? It means we are trying to find some function, let's call it L(x), such that 10^{L(x)} = x. By our starting definition, this L(x) could only be log_10(x) (since inverses are unique).

The other way, well... I'm lazy...

(edited 1 year ago)

Original post by quickquestion805

If often hear log being referred to as the inverse function of exponentials. I also hear log base a of n is equivalent to a^x=n. I don't understand how it could be both, could someone explain?

Both statements are equivalent. An exponential law essentially relates a dependent variable (y) to an independent variable (x) by an equation like $y = a^x$.

The logarithm x of a number y to base a is defined as the number to which a must be raised, or "exponentiated", to give y. That is,

$x = log_a{y} \iff y = a^x$

(Example: $1000 = 10^3$ and $log_{10}{1000} = 3$)

So there's no contradiction - your two statements are saying the same thing.

Historically, logs started out to base 10 as an aid to computation, but theoretically, and in empirical 'laws' of science, it's more often to see logs to base 'e' - so-called natural logs.

It is possible that some confusion arises because log is a binary function. You need a base and a number to get a log.

Functions such as sin are not binary, they just require one argument. Also sine and arcsine kind of sound as if they are related, even sin^-1 is ok.

Not so for log and index or power. We write a^x=b and the inverse function applied to a and b to get x is log_a(b).

Finally, these days we learn index in year 8 and log in sixth form, no wonder they don’t appear to be happy bedfellows. In the seventies, they both arose in year 8 which kind of helped I think.

Maybe we need a new notation for log and index like gradually getting rid of sin^-1 and maybe get rid of the radix symbol at the same time.

Functions such as sin are not binary, they just require one argument. Also sine and arcsine kind of sound as if they are related, even sin^-1 is ok.

Not so for log and index or power. We write a^x=b and the inverse function applied to a and b to get x is log_a(b).

Finally, these days we learn index in year 8 and log in sixth form, no wonder they don’t appear to be happy bedfellows. In the seventies, they both arose in year 8 which kind of helped I think.

Maybe we need a new notation for log and index like gradually getting rid of sin^-1 and maybe get rid of the radix symbol at the same time.

(edited 1 year ago)

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