Answer: Logarithms are mathematical functions that are used to represent the relationship between two quantities that are related by a power law. For example, if we have an exponential equation like 2^x = 8, we can use logarithms to solve for x: log2 8 = x.
In general, logarithms are written in the form logb a, where b is the base of the logarithm and a is the number we want to take the logarithm of. There are several different bases in that logarithms can be written in, including the common logarithm (base 10) and the natural logarithm (base e).
To evaluate a logarithm, we need to use the logarithmic properties and the change of base formula. One important property of logarithms is that logb (xy) = logb x + logb y. Another property is that logb (x/y) = logb x - logb y.
The change of base formula is used when we need to evaluate a logarithm in a different base than the one given. The formula is:
logb a = logc a / logc b
where c is any base other than b or a. By using this formula, we can convert a logarithm from one base to another in order to evaluate it using a calculator.
In the case of log4 25, we can use the common logarithm (base 10) or the natural logarithm (base e) to evaluate the logarithm. Using the change of base formula with the common logarithm, we have:
log4 25 = log(25) / log(4)
Using a calculator, we find that log(25) ≈ 1.39794 and log(4) ≈ 0.60206, so:
log4 25 ≈ 1.39794 / 0.60206
Simplifying this expression, we get:
log4 25 ≈ 2.32193
Rounding to the nearest thousandth, we get:
log4 25 ≈ 2.322
So, the value of log4 25 to the nearest thousandth is 2.322.
Explanation: