Evaluate the function f(x) = log3 x at x =1/27 without using a calculator
a. –4
b. –2
c. 1/27
d. –3
e. 27
so you want:
y = log3 (127)
or
3^y = 1/27 , by definition of logs
3^y = 3^-3
y = -3
f(1/27) = -3
To evaluate the function f(x) = log3 x at x = 1/27 without using a calculator, we can use the change of base formula for logarithms.
The change of base formula states that log base b of a number x can be expressed as log base c of x divided by log base c of b.
In this case, we want to evaluate log3 (1/27). Let's use base 10 as our new base, so the formula becomes:
log3 (1/27) = log(1/27) / log(3)
Now let's calculate the values without a calculator:
First, we calculate log(1/27). Since 1/27 is equivalent to 3^(-3), log(1/27) becomes log(3^(-3)). According to the logarithmic property log(b^x) = x log(b), we can rewrite this as -3 log(3):
log(1/27) = -3 log(3)
Next, we need to calculate log(3). We can estimate this value using the fact that log(10) is approximately 1. Since log(3) is a little less than log(10), we can set log(3) as 0.5:
log(3) ≈ 0.5
Now we can substitute these values back into the original equation:
log3 (1/27) = log(1/27) / log(3)
= (-3 log(3)) / log(3)
= -3 (0.5) / 0.5
= -3
Therefore, the answer is d. –3.
To evaluate the function f(x) = log3 x at x = 1/27 without a calculator, we need to remember the definition of logarithms.
The logarithm is the exponent to which a base must be raised to obtain a given number. In this case, we have the base 3.
Let's rewrite 1/27 as a power of 3:
1/27 = (1/3)^3
Now, we can use the property of logarithms that states:
log base b (b^a) = a
So, applying this property to our expression, we have:
log3 (1/3)^3 = 3 * log3 (1/3)
Next, we need to remember the property:
log base b (1/b) = -1
Using this property, we can rewrite log3 (1/3) as:
log3 (1/3) = log3 (3^-1) = -1 * log3 3 = -1
Finally, substituting this value into our expression, we have:
3 * (-1) = -3
Therefore, the value of the function f(x) = log3 x at x = 1/27 is -3.
The correct answer is (d) –3.