Problem 1. Use basic logarithmic rules to find the exact value of x if log(sqrt(x^3)) = 0.5
If `log_a b = c` , `b = a^c` . Usually if the base of the logarithm is not given, the notation log is used for the base 10.
It is given that `log_10(sqrt(x^3)) = 0.5`
`sqrt(x^3) = (x^3)^(1/2) = x^(3/2)`
Using the rule `log a^b = b*log a` ,
= `(3/2)*log_10 x`
`log_10(sqrt(x^3)) = 0.5`
=> `(3/2)*log_10 x = 0.5`
=> `log_10 x = 1/3`
=> `x = 10^(1/3)`
It is not possible to write the exact value of `10^(1/3)` as it is irrational, it is approximately equal to 2.1544346900318837217592935665194.
You have to solve `log(sqrt(x^3)) = 0.5` to get a value of x.
If log a = b, a = 10^b
`log(sqrt(x^3)) = 0.5`
or `sqrt(x^3) = 10^(0.5)`
or `x^(3/2) = 10^(1/2)`
or `x^(3/2) = (10^(1/3))^(3/2)`
or `x = 10^(1/3)`
The solution of the equation `log(sqrt(x^3)) = 0.5` is `x = 10^(1/3)`
x^3 = (10^(1/2))^2 = 10
x = 10^(1/3)
`log(sqrt(x^3)) = (1/2)`
When solving equations we just do the inverse function of the given stuff to get x by itself. (ie: We would take the root of x^2 to get x by itself)
So for this one, the inverse of logs is exponents. A log written like this is log base 10. Therefore, we will raise the whole equation by 10.
`sqrt(x^3) = 10^(1/2)`
From here, its simple equation stuff.
`x^3 = (10^(1/2))^2 = 10` (Rules of exponents)
`x = 10^(1/3)`
First rule you need to know is that if a base number is not listed you can automatically assume your base is 10. So your original can be written as `log_10(sqrt(x^3))=0.5`.
Your next rule states that log(x)=y.Basically, you can rewrite your equation to look like this: `10^(0.5)=sqrt(x^3)` . To get rid of the square root you can use the exponent 2 (you have to do this for both sides).
`(10^(0.5))^(2)=(sqrt(x^3))^(2)` which will end up simplifying to `10=x^(3)`.
That will make you final answer `x=10^((1)/(3))`.
All roots can be written as exponential fractions, so rewrite the problem as:
log x^(3/2) = .5
The third law of logs says that we can take the exponent and move it to the front, like so:
(3/2) log x = .5
From there, simplify the equation:
log x = (.5 * 2) / 3
log x = 1/3
Since no base for the log was provided, it can be safely assumed that this problem is using a base 10. With using the first rule we can rewrite the simplified form as:
10^(1/3) = x
Which is just the cube root of ten, with a calculator you find that x is equal to:
x = 2.15443469
One of the basic logarithmic rules states that log(x) = y, in which the base is understood as having a base of 10, can be rewritten as 10^y = x.
We can use this rule to solve this problem. We start off with the following equation.
log(sqrt(x^3)) = 0.5
When rearranged according to the rule mentioned above, it can be written as
10^(0.5) = sqrt(x^3)
Since an exponent of 0.5 is equivalent to taking the square root of the number, squaring both sides, or raising both sides to the second power gives us a simple equation.
10 = x^3
Last, we simply raise both sides to the (1/3) power, which the same as taking the cube root of both sides.
x = 10^(1/3) or the cube root of 10.