caligirl350 said:
can you help me to expand
Hi Caligirl:
I think you used the wrong verb. In mathematics, the verb "to expand" generally means to multiply algebraic expressions together.
You want "to evaluate" the expression log_5(0.5). In other words, you want a decimal approximation for this expression because it's the y-coordinate when x is 0.5, and this exercise is wanting you to graph some points, to get an idea of what the graph looks like for the equation y = log_5(x).
To evaluate log base-5 of 0.5, you first need a scientific calculator that has a [LOG] button or [LN] button, for evaluating base-10 or base-e logarithms, respectively. This is because most calculators do not have a base-5 button.
Using the natural logarithm [LN] button, the change-of-base formula tells us that log_5(0.5) is the same as the ratio ln(0.5)/ln(5).
In other words, we divide the natural log of the original input (0.5) by the natural log of the base (5).
Symbolically, the change-of-base formula looks like this (where I'm using b as the base):
\(\displaystyle log_{b}(Input) = \frac{ln(Input)}{ln(b)}\)
This formula works just as well, using base-10 logarithms instead of natural logarithms.
\(\displaystyle log_{b}(Input) = \frac{log(Input)}{log(b)}\)
Here's two examples of evaluating logarithms with the change-of-base formula.
\(\displaystyle log_{2}(0.9) = \frac{ln(0.9)}{ln(2)} = \frac{-0.105361}{0.693147} = -0.152003\)
\(\displaystyle log_{7}(44) = \frac{log(44)}{log(7)} = \frac{1.643453}{0.845098} = 1.944689\)
If I wrote anything that you do not understand, please reply with specific questions.
Cheers ~ Mark
