How is log(a/bc) the same as log(a) - log(b) - log(c). Should it not be '+ log (c)'?
If the fraction is a/b c then you'd have +logc
If, as I suspect, it is a/(bc) then it is indeed
loga - (logb+logc) = loga - logb - logc
Ah I get it now thank you! :)
When simplifying log expressions, it's important to understand the properties of logarithms. One of these properties is the quotient rule, which states that log(a/b) is equal to log(a) - log(b).
In the given expression, log(a/bc), we can rewrite it using the quotient rule as log(a) - log(bc).
Now, let's further simplify log(bc). Here we can apply the product rule of logarithms, which states that log(bc) is equal to log(b) + log(c).
So, substituting log(bc) in the expression log(a) - log(bc), we get:
log(a) - (log(b) + log(c))
Using the distributive property of subtraction over addition, we can rewrite it as:
log(a) - log(b) - log(c)
Thus, log(a/bc) simplifies to log(a) - log(b) - log(c). There is no '+ log(c)' term because we are subtracting log(c) in the final step.
To understand why log(a/bc) is equivalent to log(a) - log(b) - log(c), let's break it down step by step.
First, let's rewrite the expression log(a/bc) as a difference of logarithms. We know that division in the argument of a logarithm can be expressed as subtraction in the logarithm itself. Therefore, we can write log(a/bc) as log(a) - log(bc).
Next, we want to simplify log(bc). In logarithms, the product of two numbers being raised to the same power can be expressed as addition in the logarithm. Therefore, we can rewrite log(bc) as log(b) + log(c).
Now, let's substitute this back into our expression: log(a) - (log(b) + log(c)).
To remove the parentheses, we can use the property of logarithms that states that when subtracting logarithms, we can express it as the logarithm of the division of the numbers inside the parentheses. In this case, we have log(b) + log(c), so we can write it as log(b) + log(c) = log(b * c).
Substituting this back into our expression, we get log(a) - log(b * c).
Again, using the property of logarithms, we can express the division of two numbers as a difference of logarithms. Therefore, log(a) - log(b * c) can be further simplified to log(a) - (log(b) + log(c)) = log(a) - log(b) - log(c).
Therefore, we conclude that log(a/bc) is indeed equal to log(a) - log(b) - log(c), and there is no '+ log (c)' term, as you correctly questioned.
In summary, logarithm properties allow us to rewrite log(a/bc) as log(a) - log(b) - log(c), utilizing the rules of division, multiplication, and addition/subtraction within logarithms.