Math Help - a basic logarithmic inquiry...

1. a basic logarithmic inquiry...

hello--

It is understood that a logarithmic notation:

That is certainly straight forward. But, please clarify: how to correlate the above expression with the following:

10 log 2 = 3

10 log 10 = 10

10 log 1,000,000 = 60

Even entering a calculator elicits the above, but I am unable to relate "10 log 1000000 = 60" to (merely as an example) "log 2 (8) = 3"

I can do the math for "log 2 (8) = 3" (I am multiplying 2 by itself 3 times to produce 8) but I cannot explain how (again, only as an example) "
10 log 1,000,000 = 60"

I hesitate to pose this inquiry only because I feel certain the explanation will appear quite obvious...

thank you!

2. Re: a basic logarithmic inquiry...

Calculators usually do base 10 logs.

log rules say $a log_b (x) = log_b(x^a)$

so 10 log 10 or $10 log_{10} (10) = log_{10}(10^{10}) = 10$
same applies for all the other expression you provided.

basically ask your self, base b raised to what power gives the expression inside the log.

$log_2{8} = log_2{2^3}$ (2 raised to what power gives me $2^3$ ? 3, so $log_2{2^3} = 3$

3. Re: a basic logarithmic inquiry...

Originally Posted by stonewhite
hello--

It is understood that a logarithmic notation:

That is certainly straight forward. But, please clarify: how to correlate the above expression with the following:

10 log 2 = 3

10 log 10 = 10

10 log 1,000,000 = 60

Even entering a calculator elicits the above, but I am unable to relate "10 log 1000000 = 60" to (merely as an example) "log 2 (8) = 3"

I can do the math for "log 2 (8) = 3" (I am multiplying 2 by itself 3 times to produce 8) but I cannot explain how (again, only as an example) "
10 log 1,000,000 = 60"

I hesitate to pose this inquiry only because I feel certain the explanation will appear quite obvious...

thank you!
All the logarithms you mentioned above use (nearly!) the base 10.

$10 \cdot \log_{10}(10)=10~\implies~\log_{10}(10)=1~\implies ~10^1=10$

and

$10 \cdot \log_{10}(1,000,000)=60 \implies \log_{10}(1,000,000)=6 \implies \\ \\ 10^6=1,000,000$

Only

$10 \cdot \log_{10}(2)=3~\implies~\log_{10}(2)=0.3~\implies~ 10^{0.3}\approx 2$

is a very roughly rounded value.

4. Re: a basic logarithmic inquiry...

thank you both!