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, let's evaluate each one:
a) if x = -10^4, then log 10 = -4
To evaluate this statement, we need to use the definition of logarithm. The logarithm with a base of 10 tells us what exponent we need to raise 10 to in order to get the result. So, if x = -10^4, we have x = -10,000. To evaluate the logarithm of 10, we need to find an exponent that gives us 10 as a result when raised to that exponent, i.e., 10^exponent = 10. Therefore, the exponent is 1, which means log 10 = 1. Hence, statement a) is false.
b) if x = 2^8, then log 2x = 8
Using the same approach as above, if x = 2^8, then x = 256. The logarithm with a base of 2x tells us what exponent we need to raise 2x to in order to get the result. So, we need to find the exponent that gives us 2x^exponent = 2x^8 = 256. The exponent that satisfies this is 1, which means log 2x = 1. Therefore, statement b) is false.
c) log 2 2 = 4
This statement simply asks for the logarithm of 2 with a base of 2. The definition of logarithm states that the logarithm with the same base as the value being logged is always equal to 1. Therefore, log 2 2 = 1. Hence, statement c) is false.
d) if x = 3, then log 10 3 = x
If x = 3, then evaluating log 10 3 means finding the exponent that gives us 10^exponent = 3. Since there is no integer exponent that satisfies this equation, statement d) is false.
e) log 10 256 - 2log 10 a / log 10 b
This statement includes variables a and b, but it doesn't provide specific values for them. Therefore, it is not possible to determine if the statement is true or false without knowing the values of a and b.
f) log 10 (a - b) = log 10 a / log 10 b
Similar to statement e), this statement includes variables a and b without providing specific values for them. Therefore, without specific values for a and b, we cannot determine if the statement is true or false.
g) the gradient of the graph of y = 2x^x at x = 2 is 2e^e
To find the gradient of a graph at a specific point, we need to find the derivative of the function and then substitute the x-value into the derivative. We can differentiate y = 2x^x using the chain rule, which states that d/dx(f(g(x))) = f'(g(x)) * g'(x). Applying the chain rule, we get the derivative:
dy/dx = 2 * x^x * (1 + ln(x))
Substituting x = 2 into the derivative, we have:
dy/dx = 2 * 2^2 * (1 + ln(2))
dy/dx = 8 * (1 + ln(2))
dy/dx = 8 + 8ln(2)
Therefore, the gradient of the graph of y = 2x^x at x = 2 is 8 + 8ln(2), not 2e^e. Hence, statement g) is false.
h) the gradient of the graph of y = e^x at x = 2 is 2e
Again, we need to find the derivative of the function and substitute the x-value into the derivative to find the gradient. The derivative of y = e^x is simply dy/dx = e^x. Substituting x = 2 into the derivative, we get dy/dx = e^2. Therefore, the gradient of the graph of y = e^x at x = 2 is e^2, not 2e. Hence, statement h) is false.
In summary, the true statements are none of the above.