Find the value of each of the following. Leave your answers as fractions where appropriate. Do not change bases.
a) log4 (1/8)
B)10^(-10log3)<---superscript in brackets
c.2log3<-----subscript.9
d.log3<---subscript.3^2x
4 = 2^2, so 2 = 4^(1/2)
1/8 = 1/2^3 = 2^-3 = (4^1/2)^-3 = 4^(-3/2)
so, log4(1/8) = -3/2
10^(-10log3) = 10^(-log 3^10) = 10^(log 3^(-10)) = 3^(-10)
log3(9) = 2
2log3(.9) = log3(.9^2) = log3(.81)
log3(.81) = log3(81) - log3(100) = 4-log3(100)
recall that logb(b^x) = x
log3(3^2x) = 2x
To find the value of each of the given expressions, we will use the properties of logarithms and exponentiation. Let's break down each problem step by step:
a) log4(1/8):
To solve this problem, we need to find the exponent to which 4 must be raised to obtain 1/8. Remember that logarithm is the inverse operation of exponentiation. In other words, if we have log base b of x, it means b raised to the power of that logarithm will give us x.
In this case, we have log base 4 of (1/8), which means 4 raised to what power equals 1/8. So, let's rewrite the problem using exponential form:
4^x = 1/8
We can rewrite 1/8 as (1/2)^3:
4^x = (1/2)^3
Now we can see that 4 and (1/2) are reciprocals of each other, meaning their exponents will be negative counterparts:
2^2x = 2^-3
Now, we have the same base on both sides of the equation (2), so we can equate the exponents:
2x = -3
We can solve for x by dividing both sides by 2:
x = -3/2
Therefore, log4(1/8) = -3/2.
b) 10^(-10log3):
To evaluate this expression, we need to simplify it using the properties of logarithms and exponentiation.
We can rewrite 10 as a power of 10 with a base of 3 by using a logarithm:
10 = 3^(log3(10))
Now, we can rewrite the expression as:
3^(log3(10))^(-10log3)
According to the exponentiation property, when we raise a power to another power, we multiply the exponents. Applying this property, we get:
3^(-10 * log3(10))
Since we have the same base (3), we can drop the base and equate the exponents:
-10 * log3(10) = -10
So, 10^(-10log3) simplifies to 10^(-10).
c) 2log3.9:
In this expression, we have a logarithm with a subscript of 9. This means the logarithm is base 9.
To evaluate this expression, we need to rewrite it using exponential form:
2^(log9(3))
Now, we can rewrite it as:
9^(log9(3))^2
Since the base is the same (9), we can drop the base and equate the exponents:
(log9(3))^2 = 1
So, 2log3.9 simplifies to 2.
d) log3(3^2x):
In this expression, we have a logarithm with a subscript of 3^2x. This means the logarithm is base 3^2x.
To simplify this expression, we can use the logarithm property that states log base b of b^x is equal to x.
Applying this property, we have:
log3(3^2x) = 2x
Therefore, log3(3^2x) simplifies to 2x.