use the properties of logarithms to expand the expression as sum, difference or multiple of logarithms. Simplify.
a.log(x^3/2)
b.log base4 16a
log (x^3/2)=
logx^3 - log 2
3logx-log2
log4(16a)=log4(4)+log4(a)
= 2+log4(a)
a. To expand the expression a.log(x^3/2) using the properties of logarithms, we can apply the power rule:
log(x^3/2) = (3/2)log(x)
So, a.log(x^3/2) can be rewritten as (3/2)a.log(x).
b. To expand the expression log base 4 of 16a, we can use the change of base formula:
log base 4 of 16a = (log base a of 16) / (log base a of 4)
Now, let's calculate the logarithms:
log base a of 16 = log base a of (2^4) = 4log base a of 2
log base a of 4 = log base a of (2^2) = 2log base a of 2
Therefore, log base 4 of 16a = (4log base a of 2) / (2log base a of 2)
Since log base a of 2 appears in both the numerator and denominator, we can cancel them out, resulting in:
log base 4 of 16a = 4 / 2
Simplifying further:
log base 4 of 16a = 2
a. To expand the expression log(x^(3/2)), we can use the property of logarithms that states log(a/b) = log(a) - log(b).
Applying this property, we can rewrite the expression as:
log(x^(3/2)) = log(x^3) - log(2)
Now we can simplify it further:
log(x^3) - log(2)
b. To expand the expression log base 4 of 16a, we can use the definition of logarithms to rewrite it in exponent form.
In exponent form, log base b of a = c is equivalent to b^c = a.
Applying this definition, we can write the expression as:
4^c = 16a
To find the value of c, we need to determine the exponent that allows 4 to equal 16. By observing that 4^2 = 16, we conclude that c = 2.
Therefore, we can rewrite the expression as:
log base 4 of 16a = 2
Hence, the expression log base 4 of 16a can be expanded and simplified to 2.
a) To expand the expression a.log(x^3/2), we can use the property of logarithms that says log(a*b) = log(a) + log(b). In this case, we have a coefficient "a" multiplied by log(x^(3/2)).
Using the property mentioned above, we can rewrite the expression as:
log((x^(3/2))^a)
Now, another property of logarithms states that log(a^b) = b * log(a). Applying this property, we get:
log(x^(3/2*a))
Finally, we can simplify the expression to:
3/2 * a * log(x)
So, the expanded form of a.log(x^3/2) is 3/2 * a * log(x).
b) To expand the expression log base 4 of 16a, we can use the change of base formula for logarithms. The change of base formula states that log base b of a equals log base c of a divided by log base c of b. In this case, we want to change the base from 4 to a more common logarithm base, such as 10 or natural logarithm (ln).
Using the change of base formula, we can rewrite the expression as:
log base 4 of 16a = log base 10 of 16a / log base 10 of 4
Now, we know that log base 10 of 16 (or any power of 10) is simply the exponent:
log base 10 of 16 = 2
And log base 10 of 4 is equal to 1:
log base 10 of 4 = 1
Therefore, the expression can be simplified to:
log base 4 of 16a = 2 / 1 = 2
So, the expanded form of log base 4 of 16a is simply 2.