which three statements are true?

a) if x= -10^4 then log 10 = -4
b)if x= 2^8 then log 2x = 8
c) log2 2= 4
d) if x= 3 then log10 3=x
e) log 10 256-2log 10 a/log 10 b
f)log 10 (a-b)= log 10 a/log 10 b
g) the gradient of the graph of y= 2x^x at x= 2 is 2e^e
h) the gradient of the graph of y= e^x at x= 2is 2e

To determine which statements are true, we need to evaluate each statement individually. Let's go through them one by one:

a) if x = -10^4 then log 10 = -4
To check if this statement is true, we substitute x = -10^4 into the equation.
log 10^x = log 10^(-10^4) = log 10^(-10,000).
Recall that log a^b = b * log a.
Therefore, log 10^(-10,000) = -10,000 * log 10.
Using the definition that log 10 = 1, we have:
-10,000 * 1 = -10,000.

Thus, the statement a) if x = -10^4 then log 10 = -4 is false.

b) if x = 2^8 then log 2x = 8
To verify this statement, we substitute x = 2^8 into the equation.
log 2x = log 2(2^8) = log 2^9.
Since log a^b = b * log a, we have:
log 2^9 = 9 * log 2.
If we express 2^9 as 512, we have:
9 * log 2 = 9 * 0.301 = 2.709.

Therefore, the statement b) if x = 2^8 then log 2x = 8 is false.

c) log2 2 = 4
This statement is asking if log base 2 of 2 is equal to 4.
Since 2^4 = 16, we can write the equation as:
log2 16 = 4.
Using the definition of logarithm, we have:
2^4 = 16, which verifies that the statement is true.

Thus, the statement c) log2 2 = 4 is true.

d) if x = 3 then log10 3 = x
This statement claims that if x = 3, then log base 10 of 3 is equal to x.
Substituting x = 3 into the equation, we have:
log10 3 = 3.
Using the definition of logarithm, we have:
10^3 = 3, which is not true.

Therefore, the statement d) if x = 3 then log10 3 = x is false.

e) log 10 256 - 2log 10 a / log 10 b
This statement appears to be incomplete. To determine its truth value, we need more information or clarification.

f) log 10 (a - b) = log 10 a / log 10 b
This statement claims that the logarithm of (a - b) with base 10 is equal to the logarithm of a divided by the logarithm of b with base 10.
Using logarithm properties, we can rewrite the equation as:
log 10 (a - b) = log 10 a - log 10 b.
This logarithmic equation is true and holds for any values of a and b.

Therefore, the statement f) log 10 (a - b) = log 10 a / log 10 b is true.

g) the gradient of the graph of y = 2x^x at x = 2 is 2e^e
To determine if this statement is true, we need to calculate the gradient (or derivative) of the function at x = 2 and compare it to 2e^e.
Differentiating y = 2x^x using the power rule, we get:
dy/dx = 2x^x * (1 + ln(x)).
Evaluating the derivative at x = 2, we have:
dy/dx = 2(2^2) * (1 + ln(2)) = 8(1 + ln(2)).
Calculating 2e^e, we get:
2e^e = 2 * e^(2.718) ≈ 2 * 15.154 ≈ 30.308.

Therefore, the statement g) the gradient of the graph of y = 2x^x at x = 2 is 2e^e is false.

h) the gradient of the graph of y = e^x at x = 2 is 2e
To verify this statement, we need to calculate the gradient (or derivative) of the function at x = 2 and compare it to 2e.
Differentiating y = e^x using the chain rule, we get:
dy/dx = e^x * 1 = e^x.
Evaluating the derivative at x = 2, we have:
dy/dx = e^2.
Calculating 2e, we get:
2e ≈ 2 * 2.718 ≈ 5.436.

Therefore, the statement h) the gradient of the graph of y = e^x at x = 2 is 2e is false.

In summary, the true statements are c) log2 2 = 4 and f) log 10 (a - b) = log 10 a / log 10 b.