1) True/false: For any real number x,

ln e^x = x.
I think that is true.

2)true/false:
(2 ln x)/ ln 10 = ln x^2 - ln 10 for all positive x.
I think this is true too.

1) True. That's the definition of the natural log.

2) No.
ln(x^2/10) = 2lnx - ln 10

That's not the same as 2lnx/ln10

ln x2/ln 10 = log10x^2

1) True/false: For any real number x, ln e^x = x.

True.

Explanation: The natural logarithm function, ln(x), is the inverse of the exponential function, e^x. This means that ln(e^x) will give us the value of x. Therefore, ln(e^x) = x for any real number x.

2) True/false: (2 ln x)/ ln 10 = ln x^2 - ln 10 for all positive x.

False.

Explanation: Let's simplify both sides of the equation to see if they are equal.

On the left side, we have (2 ln x) / ln 10. We can use the property of logarithms to rewrite this as ln(x^2) / ln 10.

On the right side, we have ln x^2 - ln 10. Using the subtraction property of logarithms, we can rewrite this as ln(x^2 / 10).

Therefore, the equation becomes ln(x^2) / ln 10 = ln(x^2 / 10).

Here's where we can see that the equation is false. It is not true that ln(x^2) / ln 10 is equal to ln(x^2 / 10) for all positive x. Therefore, the statement is false.

1) True/false: For any real number x, ln e^x = x.

This statement is true.
To explain why, we need to understand the properties of logarithms and the number e.
The natural logarithm (ln) is the inverse function of the exponential function with base e. The number e is a mathematical constant approximately equal to 2.71828.

To simplify ln e^x, we can use the fact that e^x is the base e exponential function.
So ln e^x is equivalent to saying "What exponent do I need to raise e to in order to get e^x?" And the answer is simply x.

Therefore, ln e^x = x is true for any real number x.

2) True/false:
(2 ln x) / ln 10 = ln x^2 - ln 10 for all positive x.

This statement is true.
To explain why, we need to use the properties of logarithms and algebraic manipulation techniques.

First, let's simplify each side of the equation step by step:

Left-hand side: (2 ln x) / ln 10.
Using the property of logarithms, ln a - ln b can be simplified to ln(a/b):
(2 ln x) / ln 10 = ln(x^2) / ln 10
By the power property of logarithms, ln(a^b) can be simplified to b ln a:
ln(x^2) / ln 10 = 2 ln x / ln 10

Right-hand side: ln x^2 - ln 10.
Using the property of logarithms, ln a - ln b can be simplified to ln(a/b):
ln x^2 - ln 10 = ln(x^2 / 10)

Now, let's compare the two sides of the equation:
2 ln x / ln 10 = ln(x^2 / 10)

To show that these two expressions are equal for all positive x, we can simplify further:

Using the property of logarithms, ln a - ln b can be simplified to ln(a/b):
ln(x^2 / 10) can be written as ln(x^2) - ln(10)

Now we have:
2 ln x / ln 10 = ln(x^2) - ln(10)

By comparing the two expressions, we can see that the left-hand side is equal to the right-hand side:

2 ln x / ln 10 = ln(x^2) - ln(10)

Therefore, the statement (2 ln x) / ln 10 = ln x^2 - ln 10 for all positive x is true.