logarithm question

[gkagawa]

I don't understand the principle behind logarithms as exponents. The question is: f(x)=10[sup:15v2h2v9]x[/sup:15v2h2v9] and g(x)=logx. State the domain and range of f(g(x)).
So I got : f(g(x))= 10[sup:15v2h2v9]logx[/sup:15v2h2v9]. I was told the answer for the domain was x, but can you explain why?

 

[tkhunny]

Logarithms did not start their existence as exponents. It took a good 50-100 before they discovered their true nature. Don't be too stressed if it takes you a week or two.

Think on the integers...

10^1 = 10
log(10) = 1

10^2 = 100
log(100) = 2

10^3 = 1000
log(1000) = 3

Think on g(f(x)) = log(10^x)

Using rules of logarithms: g(f(x)) = x*log(10) = x*1 = x

 

[lookagain]

I don't understand the principle behind logarithms as exponents. The question is: f(x)=10[sup:2c4tjedp]x[/sup:2c4tjedp] and g(x)=logx. State the domain and range of f(g(x)).
So I got : f(g(x))= 10[sup:2c4tjedp]logx[/sup:2c4tjedp]. I was told the answer for the domain was x, but can you explain why?

\(\displaystyle f(g(x)) \ne x\)

Although there is no restriction on x for 10^x, the domain of g(x) is x > 0, or (0, oo).
So the domain of f(g(x)) is x > 0, or (0, oo).

As x > 0 for the domain, then the range of f(g(x)) is x > 0, or (0, oo).

\(\displaystyle f(g(x)) = x, \ x > 0\)

 

[gkagawa]

OK, I understand the answer you gave where g(f(x))=x, but I am still unclear about how the log x part can be an exponent to the base10. Is there another way you can rewrite the equation f(g(x))=10[sup:2jmvyq48]logx[/sup:2jmvyq48] and manipulate it so that I can see the light? Would it be correct to state that f(g(x))= log(log(x))? If this is true, can I state that f(g(x)) = x?

 

[tkhunny]

Whoever told you "the Domain is x" needs to reread the definition of "Domain". It will NEVER take a form like that.

Forget that log(log(x)) experiment. You are confusing yourself.

These are equivalent expressions for Real Numbers a, b, and c (a, b > 0):

\(\displaystyle log_{b}(a)\;=\;c \iff b^{c}\;=\;a\)

Memorize it as a rule. Learn it as a transformation. Invite it over for dinner. Make it part of your life.