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
Things you should know:
Log(A)-Log(B) = Log(A/B).
Log(A)+Log(B) = Log(A*B).
Log(A)^n = n*Log(A).
Note: When no base is given, it is assumed to be 10.
N^(1/2) = sqrt(N).
Step-by-step:
2(Log6(15)-Log6(5) + 1/2Log6(1/25).
2(Log6(15/5)) + Log6(1/25)^(1/2) =
2(Log6(3)) + Log6(sqrt(1/25)) =
Log6(3^2) + Log6(1/5) =
Log6(9) + Log6(1/5) =
Log6(9*1/5) =
Log6(9/5).
To condense the given expression, we will use the properties of logarithms. Here are the step-by-step instructions:
Step 1: Simplify the subtraction inside the parentheses.
We can use the quotient rule of logarithms, which states that log base b of (a/c) can be rewritten as log base b of a minus log base b of c.
So, we can rewrite the expression as
2(log base 6 of 15/5) + 1/2 log base 6 of 1/25.
Step 2: Simplify the division inside the logarithm.
In this case, 15/5 simplifies to 3.
So, the expression becomes 2(log base 6 of 3) + 1/2 log base 6 of 1/25.
Step 3: Apply the power rule of logarithms.
According to the power rule, we can bring the coefficient in front of the logarithm as the exponent inside the logarithm.
Using this rule, we can rewrite the expression as
log base 6 of 3^2 + log base 6 of (1/25)^(1/2).
Step 4: Simplify the exponents.
3^2 simplifies to 9, and (1/25)^(1/2) simplifies to 1/5.
So, the expression simplifies to
log base 6 of 9 + log base 6 of 1/5.
Step 5: Apply the product rule of logarithms.
The product rule states that log base b of (mn) can be rewritten as log base b of m plus log base b of n.
Using this rule, we can rewrite the expression as
log base 6 of (9 * 1/5).
Step 6: Simplify the multiplication inside the logarithm.
9 * 1/5 simplifies to 9/5.
So, the expression becomes
log base 6 of 9/5.
Step 7: Condense the expression.
Finally, we have condensed the expression to
log base 6 of 9/5.
And that's the condensed form of the given expression.