If log_10(2)=x and log_10(3)=y, then what is the value of log_10(15) in terms of x and y?
*the first logarithmic equation reads: the logarithm of x with base 10 is is 2.
log_10(15)=log(15)/log(10)
log(10)=1
log(10)=log(5*2)=log(5)+log(2)=1
log(2)=x
so replacing it is log(5)=1-x
then in the first part
log(15)=log(3*5)=log(3)+log(5)=y+1-x
so log_(10)=log(15)/log(10)=y+1-x/1
so finally it equals log_10(15)=y+1-x
hope you understand my solution, I didn't put the base 10 in some parts of the exercise because it is understandable
or
15 = (10x3÷2)
take log of both sides
log 15 = log (10x3÷2)
= log10 + log3 - log2
= 1 + y - x
To find the value of log_10(15) in terms of x and y, we need to use logarithmic properties.
We know that:
log_10(15) = log_10(3 * 5)
Using the property log_a(b * c) = log_a(b) + log_a(c), we can rewrite the equation as:
log_10(15) = log_10(3) + log_10(5)
Since we are given the values of log_10(2) (x) and log_10(3) (y), we can substitute them into the equation:
log_10(15) = y + log_10(5)
However, we are not given the value of log_10(5) directly. To find this, we need to use logarithmic properties again.
We know that:
log_10(5) = log_10(10/2)
Using the property log_a(b/c) = log_a(b) - log_a(c), we can rewrite the equation as:
log_10(5) = log_10(10) - log_10(2)
Since we are given the value of log_10(2) (x), we can substitute it into the equation:
log_10(5) = log_10(10) - x
Now, we can substitute this value back into the previous equation:
log_10(15) = y + (log_10(10) - x)
Simplifying further, we get:
log_10(15) = y - x + log_10(10)
Therefore, the value of log_10(15) in terms of x and y is y - x + log_10(10).