To find the zeros of the function f(x) = (2^x - 1)(-3^x + 1), we need to find the values of x for which f(x) equals zero.
First, let's find the first zero using the given equation:
x = (ln2 + ln3)/(ln2 - ln3)
To simplify this, we can use the logarithmic properties. The equation can be written as:
x = ln(2 * 3) / ln(2 / 3)
x = ln(6) / ln(2/3)
To evaluate this expression, we need to use the logarithmic rules. The numerator can be simplified to ln(6), and the denominator can be rewritten using the rule: ln(a/b) = ln(a) - ln(b). Hence, we have:
x = ln(6) / (ln(2) - ln(3))
Similarly, we can apply the same logarithmic rules to the second given equation:
x = (ln2 - ln3)/(ln2 + ln3)
x = ln(2) - ln(3) / ln(2) + ln(3)
Now, let's simplify the third equation:
x = 2 / (ln2 - ln3)
Finally, let's evaluate the fourth equation:
x = -ln(5) / ln(1)
Since ln(1) equals zero, we have:
x = -ln(5) / 0
However, division by zero is undefined, so this equation does not have a valid solution.
Moving on to the second question, solving the equation log(27)[log(x)10] = 1/3, we can follow these steps:
Start by rewriting the equation in an equivalent exponential form:
27^(1/3) = log(x)10
Since 27^(1/3) can be simplified to 3, the equation becomes:
3 = log(x)10
To remove the logarithm, we can rewrite this equation using the definition of logarithms:
10^3 = x
Simplifying further, we have:
x = 1000
So, the solution to the equation log(27)[log(x)10] = 1/3 is x = 1000.
Regarding your clarification, when you mention "(inverse)" in the context of square roots, it seems you are referring to the radical symbol (√) over the number. However, the notation you provided (e.g., 3(inverse)sqrt10) is not clear. It would be helpful to use commonly understood mathematical notation to communicate more effectively.