The expression 1/3 log (a)-3 log(b) is equivalent to_________, i know the answer but i don't know how to get it.
Use your rules of logs
Log a^(1/3) - log b^3
Log (a^(1/3)) / (b^3)
To simplify the expression 1/3 log(a) - 3 log(b), we can use the properties of logarithms and algebraic operations. Here's how you can do it step-by-step:
Step 1: Apply the power rule of logarithms.
The power rule states that log(base b)(x^n) = n * log(base b)(x).
In this case, we have:
1/3 log(a) = log(a^(1/3))
So, the expression becomes:
log(a^(1/3)) - 3 log(b)
Step 2: Simplify the exponent.
Since the exponent of 1/3 means taking the cube root, we can rewrite it as:
log((a^(1/3))^3) - 3 log(b)
This simplifies to:
log(a) - 3 log(b)
Step 3: Apply the quotient rule of logarithms.
The quotient rule states that log(base b)(x/y) = log(base b)(x) - log(base b)(y).
Applying the quotient rule to our expression, we get:
log(a) - 3 log(b) = log(a) - log(b^3)
Final Answer:
The simplified expression is log(a) - log(b^3).
Note: It's important to mention that if you have specific values for 'a' and 'b', you can further simplify the expression. However, without specific values, this is the most simplified form.
To simplify the expression 1/3 log(a) - 3 log(b), we can use some logarithmic properties and rules.
First, let's rewrite the expression using the logarithmic rule that states:
log(base b) (x^y) = y * log(base b) (x)
Using this rule, we have:
1/3 log(a) - 3 log(b) = log(a^(1/3)) - log(b^3)
Next, we can apply another logarithmic rule, which says that the difference of two logarithms is equal to the logarithm of the quotient of their arguments:
log(a) - log(b) = log(a / b)
Using this rule, we get:
log(a^(1/3)) - log(b^3) = log(a^(1/3) / b^3)
Now, we can simplify the expression inside the logarithm by using the rule for dividing powers with the same base:
a^(m/n) / b^(p/q) = (a^m / b^p)^(1/nq)
Applying this rule, we have:
log(a^(1/3) / b^3) = log((a / b^9)^(1/3))
Since raising something to the power of 1/3 is equivalent to taking the cube root, we have:
log((a / b^9)^(1/3)) = log(cbrt(a / b^9))
Therefore, the simplified form of the expression 1/3 log(a) - 3 log(b) is log(cbrt(a / b^9)).