True or false:
2 ln x / ln 5 = ln x^2 - ln 5
for all positive values of x.
False.
All we need is one counterexample.
let x = e
LS = 2 ln2/ln5 = 2/ln5
RS = ln e^2 - ln5
= 2lne - ln5
= 2 - ln5
≠ LS
2 lnx/ln 5 = 2ln x - ln 5
only for lnx = ln^2(5)/(2 ln 5 - 2)
To determine whether the statement is true or false, we need to simplify both sides of the equation and check if they are equal.
Starting with the left-hand side (LHS) of the equation:
2 ln x / ln 5
First, we can use the logarithmic identity ln(a) - ln(b) = ln(a/b) to rewrite 2 ln x / ln 5 as ln(x^2) / ln 5:
ln(x^2) / ln 5
Now, let's simplify the right-hand side (RHS) of the equation:
ln x^2 - ln 5
Using a similar logarithmic identity ln(a) + ln(b) = ln(a * b), we can rewrite ln x^2 - ln 5 as ln(x^2/5):
ln(x^2/5)
Now that we have the simplified versions of both sides, the equation becomes:
ln(x^2) / ln 5 = ln(x^2/5)
To determine if this equation is true for all positive values of x, we need to check if both sides are equal.
Taking the natural logarithm (ln) of both sides won't help us reach a conclusion since we have two different logarithms in the equation. Instead, we can simplify further by using the property of logarithms that states if ln(a) = ln(b), then a = b.
Therefore, to check if the equation is true, we need to equate the arguments of the logarithms:
x^2 / 5 = x^2/5
Now, we can see that both sides of the equation are equal. Therefore, the statement "2 ln x / ln 5 = ln x^2 - ln 5" is true for all positive values of x.
To summarize, the statement is true, and we determined this by simplifying both sides of the equation and checking if they are equal.