Can you please explain step by step on how to do this problem?
Condense the expression.
2(log base 6 of 15 - log base 6 of 5) + 1/2 log base 6 of 1/25
2 log6 (3) + .5 log6 (1/25)
log6 (3^2) + log6 ((1/25)^.5)
log6 (9) + log6 (1/5)
log6 (9/5)
Sure! To condense the expression 2(log base 6 of 15 - log base 6 of 5) + 1/2 log base 6 of 1/25, we can use the logarithmic properties to simplify it.
Step 1: Apply the quotient rule of logarithms
The quotient rule states that log base a of (b / c) is equal to log base a of b minus log base a of c.
In this case, we have log base 6 of 15 and log base 6 of 5 as the numerator and denominator of the first logarithm, respectively. Applying the quotient rule, we get:
2(log base 6 of 15 - log base 6 of 5) + 1/2 log base 6 of 1/25
= 2(log base 6 of (15/5)) + 1/2 log base 6 of 1/25
Simplifying further, we have:
= 2(log base 6 of 3) + 1/2 log base 6 of 1/25
Step 2: Simplify the log base 6 of 1/25
To simplify log base 6 of 1/25, we can use the exponent rule of logarithms, which states that log base a of b^c is equal to c times log base a of b.
In this case, we have log base 6 of 1/25. Using the exponent rule, we can rewrite it as:
1/2 log base 6 of 1/25
= 1/2(log base 6 of (1/5)^2)
= 1/2(2 log base 6 of 1/5)
Simplifying further, we have:
= log base 6 of 1/5
Step 3: Combine the logarithms
Now that we have simplified both parts of the expression, we can combine them. Plugging the simplified log base 6 of 3 and log base 6 of 1/5 back into the original expression, we get:
2(log base 6 of 3) + 1/2 log base 6 of 1/25
= 2 log base 6 of 3 + log base 6 of 1/5
And there you have it! The expression 2(log base 6 of 15 - log base 6 of 5) + 1/2 log base 6 of 1/25 can be condensed to 2 log base 6 of 3 + log base 6 of 1/5.