can anyone please explain base changing theorum of log.
Answers
Step-by-step explanation:
We can change the base of any logarithm by using the following rule:
Notes:
When using this property, you can choose to change the logarithm to any base \greenE xxstart color #0d923f, x, end color #0d923f.
As always, the arguments of the logarithms must be positive and the bases of the logarithms must be positive and not equal to 111 in order for this property to hold!
Example: Evaluating \log_2(50)log
2
(50)log, start base, 2, end base, left parenthesis, 50, right parenthesis
If your goal is to find the value of a logarithm, change the base to 101010 or eee since these logarithms can be calculated on most calculators.
So let's change the base of \log_2(50)log
2
(50)log, start base, 2, end base, left parenthesis, 50, right parenthesis to {\greenD{10}}10start color #1fab54, 10, end color #1fab54.
To do this, we apply the change of base rule with b=2b=2b, equals, 2, a=50a=50a, equals, 50, and x=10x=10x, equals, 10.
\begin{aligned}\log_\blueD{2}(\purpleC{50})&=\dfrac{\log_{\greenD{10}}(\purpleC{50})}{\log_{\greenD{10}}(\blueD2)} &&\small{\gray{\text{Change of base rule}}}\\ \\\\\\ &=\dfrac{\log(50)}{\log(2)} &&\small{\gray{\text{Since} \log_{10}(x)=\log(x)}} \end{aligned}
log
2
(50)
=
log
10
(2)
log
10
(50)
=
log(2)
log(50)
Change of base rule
Sincelog
10
(x)=log(x)
We can now find the value using the calculator.
\begin{aligned}\phantom{\log_2(50)}\approx 5.644 \end{aligned}
log
2
(50)≈5.644
I'd like to see this example done with base-e instead.