Help Understanding What a Logarithm IS

[Moonseed]

I need help defining a logarithm.

My book simply says: A logarithm is an exponent.

This stumped me because I can't see how that is. I don't know what question to ask, but I might not be apprehending the relationship between an expo. function and a log. function.

 

[d_leet]

I need help defining a logarithm.

My book simply says: A logarithm is an exponent.

This stumped me because I can't see how that is. I don't know what question to ask, but I might not be apprehending the relationship between an expo. function and a log. function.

Well the logarithm base b is defined as

y = log_b(x)

if

x = b^y

Notice that in the second equation y is the exponent, and in the first equation it is the logarithm, thus since these two equations are identical and express the exact same mathematical relationship it follows that a logarithm is essentially an exponent.

 

[Moonseed]

Thanks, that helped a lot.

I'm curious about why 'x' ended up where it is when the logarithm was converted to an exponential function or vice versa.

 

[Ateowa]

When converting from an exponential to a log (We'll use x=b^y as an example), you bring the y down. This bumps the b down into the base (Making it x=_by). Then you switch the x and the y (Finishing the conversion to y=log_bx). Did that make sense?

 

[HallsofIvy]

In technical terms, the logarithm is the "inverse" of the exponential function. That is, you swap the "x" and "y" values: if y= f(x) then x= f^-1(y). As Ateowa said, if y= log_ax then x= a^y.