which 3 are correct

a) if x= -10^4 then log10 x = -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 16=0
f)log 10 (a-b)= log 10 a/log 10 b
g) the gradient of the graph of y= 2e^x at x= 2 is 2e^2
h) the gradient of the graph of y= e^x at x= 2is 2e

What is your thinking on this?

To determine which statements are correct, we can evaluate each statement one by one:

a) if x = -10^4, then log10 x = -4:
To answer this, we need to apply the logarithmic property that states log10 x = y if and only if 10^y = x. In this case, 10^(-4) ≠ -10^4, so statement a) is incorrect.

b) if x = 2^8, then log2x = 8:
Similar to the previous statement, we need to apply the logarithmic property log2x = y if and only if 2^y = x. Since 2^8 = 256, statement b) is correct.

c) log2 2 = 4:
The logarithm log2 2 = y can be rewritten as 2^y = 2. Since 2^1 = 2, statement c) is incorrect.

d) if x = 3, then log10 3 = x:
To determine if this statement is correct, we can calculate log10 3 and check if it is equal to x. log10 3 ≈ 0.477, which is not equal to 3, so statement d) is incorrect.

e) log10 256 - 2log10 16 = 0:
To evaluate this statement, we can use the logarithmic property log a - log b = log(a/b). Using this property, we can rewrite the expression as log10 (256/16^2). Simplifying further, 256/16^2 equals 256/256 = 1. Hence, log10 1 = 0. Therefore, statement e) is correct.

f) log10 (a - b) = log10 a / log10 b:
To determine if this statement is correct, we need to use the logarithmic property that states log a - log b = log(a/b). Since log10 (a - b) cannot be simplified further, statement f) is incorrect.

g) the gradient of the graph of y = 2e^x at x = 2 is 2e^2:
To find the gradient of the graph, we need to differentiate the function y = 2e^x with respect to x. The derivative of e^x is e^x, so the gradient at any point is 2e^x. Plugging in x = 2, the gradient becomes 2e^2. Therefore, statement g) is correct.

h) the gradient of the graph of y = e^x at x = 2 is 2e:
Similar to statement g), we differentiate the function y = e^x with respect to x. The derivative of e^x is e^x, so the gradient at any point is e^x. Plugging in x = 2, the gradient becomes e^2. Therefore, statement h) is incorrect.

Based on the above evaluations:

Correct statements: b), e), g).
Incorrect statements: a), c), d), f), h).