Which of the following is equivalent to log 8 16 ?
To determine which expression is equivalent to log 8 16, we need to understand the logarithm function. The logarithm is the inverse function of exponentiation. It helps us solve for the exponent when a base is raised to a certain power.
In this case, we have the expression log 8 16. This represents the logarithm of 16 with base 8. It asks for the exponent to which 8 must be raised to give 16.
We can use the change of base formula to make it easier to evaluate. The change of base formula states that any logarithm with the base 'a' can be expressed as the logarithm with any other base 'b' divided by the logarithm of 'a' with the base 'b'.
Using this formula, we can convert log 8 16 to log 16 / log 8.
Now, we need to calculate the logarithm of 16 with base 8 and the logarithm of 8 with base 8.
The logarithm of 16 with base 8 can be found by asking: "To what power must 8 be raised to get 16?". Evaluating this, we find that 8^2 = 64 and 8^3 = 512. Since 16 is between these two numbers, we can approximate that 8^2.5 ≈ 16. Therefore, log 8 16 = 2.5.
Similarly, the logarithm of 8 with base 8 is 1. We raise the base 8 to the power of 1 to get the value of 8.
Now, we can substitute these values back into our expression:
log 16 / log 8 = 2.5 / 1 = 2.5
Therefore, the expression log 8 16 is equivalent to 2.5.
To find the equivalent expression for log base 8 of 16, we need to determine the exponent that 8 needs to be raised to in order to get 16.
So, let's write the equation as:
8^x = 16
To solve for x, we can take the logarithm of both sides of the equation. In this case, the base of the logarithm is 8:
log 8 (8^x) = log 8 (16)
Since the base of the logarithm matches the base of the exponent, the logarithm and exponent cancel each other out, and we are left with:
x = log 8 (16)
Therefore, log 8 16 is equivalent to x.