Why log x = exponent; rather than log f(x) = exponent, if f(x) = 10^x?

Question

Terminology question: I find the switching of x and y a contradiction, since b^x=y, or 10^x=y, or e^x=y or f(x). Why b^y=x when the function is the base to an exponent equals the y value or f(x)?

 

 

 

Thanks,

Richard Ackroyd

Answer 1

if f(x) = 10^x then in this function you have base ten, exponent x.

Therefore log f(x) = x.

Since y = f(x), in general, this is, equivalently, log y = x.

 

That's all there is, by definition. LOGARITHMS ARE EXPONENTS.

 

you are familiar with powers of ten: 10^-1 = 0.1, 10⁰ =1, 10¹ = 10, 10² = 100 and even 10^½ = 3.162 (sq root of ten); 

therefore log 1 = 0, log (1/10) = -1, log 100 = 2, log 3.162 = 0.5.

 

Furthermore 10^0.30103 = 2 so log 2 = 0.30103.

 

Soon you should be learning about natural logarithms.  Any more questions?

Answer 2

If f(x) = 10^x, then by the properties of logarithms

 

log[f(x)] = log(10^x) = x log10 = x(1) = x

 

log[f(x)] = x